Excel ms365, 198 bytes

=LET(f,REGEXREPLACE,g,SORTBY,s,g(A.:.A,-LEN(A.:.A)),d,REDUCE(,s,LAMBDA(x,y,TOCOL(IF(REGEXTEST(x,y),x,f(HSTACK(x&0&y,y&0&x),"(.*)0\1","$1")),3))),REDUCE(@g(d,LEN(d)),s,LAMBDA(a,b,f(a,b,"\u$0",1,1))))

Perl 5 -Mntheory=:all -p, 120 bytes

$x=$_;forperm{$_='';for$s(@a[@_]){$S=$s;/$/;s/.{$+[0]}\K/chop$S/e until s/$s/\u$&/i}$_=$x=length$x>y///c?$_:$x}@a=/\w+/g

Try it online!

Looks like it's waste of time to try to avoid checking all permutations.

Input is lowercase, single line, with any non-word separators between words. More readable:

$x = $_;
forperm {
    $_ = '';
    for $s ( @a[ @_ ]){
        $S = $s;
        /$/;
        s/.{$+[0]}\K/chop$S/e until s/$s/\u$&/i
    }
    $_ = $x = length $x > y///c ? $_ : $x
} @a = /\w+/g

JavaScript (ES6), 135 bytes

Expects a list of words in title case.

f=(a,p=o=0)=>a.map(w=>f(a.filter(W=>W!=w),(p+0+w).replace(/(.?(.*))(?:(.*)0\1$|0\1)/i,(_,a,b,c)=>[w[-!a]]+b+[c])))+a||p[o.length]?o:o=p

Try it online!

Method

For each permutation \$[w_1,w_2,\dots,w_n]\$ of the input list:

• we start with an empty string \$p\$
• for each word \$w_i\$, we do \$p\gets p\oplus w_i\$

Where the operator \$\oplus\$ will either:

• If a full occurrence of \$w_i\$ already exists anywhere in \$p\$ (case insensitive):

  → capitalize the leading letter of \$w_i\$ in \$p\$

  Example: \$``\text{Clocks"}\oplus``\text{Lock"}=``\text{C}\color{red}{\text{L}}\text{ocks"}\$

• If \$p\$ ends with a prefix of \$w_i\$ (case insensitive):

  → capitalize the leading letter of \$w_i\$ in \$p\$ and append the missing letters

  Example: \$``\text{Fish"}\oplus``\text{Shrimp"}=``\text{Fi}\color{red}{\text{S}}\text{h}\color{red}{\text{rimp"}}\$

• Otherwise:

  → append \$w_i\$ to \$p\$ (standard concatenation)

  Example: \$``\text{Cat"}\oplus``\text{Dog"}=``\text{Cat}\color{red}{\text{Dog}}\text{"}\$

The output is the shortest \$p\$.

Implementation of \$\oplus\$

(p + 0 + w)                    // we work on the concatenation of p, "0" and w
.replace(                      //
  /(.?(.*))(?:(.*)0\1$|0\1)/i, // /(.?(.*)) capture a reference substring in p
                               //           with the leading letter apart
                               // (?:       followed by either:
                               //   (.*)      an optional padding string,
                               //   0         the separator,
                               //   \1        the reference substring,
                               //   $         the end of the string
                               //             (i.e. the reference substring is
                               //             a full occurrence of w)
                               // |         or:
                               //   0         the separator,
                               //   \1        the reference substring
                               //             (i.e. the reference substring is
                               //             a suffix of p and a prefix of w)
                               // )/i       case insensitive
  (                            //
    _,                         // the full match is ignored
    a,                         // a = ref. substring
    b,                         // b = ref. substring without its leading letter
    c                          // c = padding string
  ) =>                         //
    [w[-!a]] +                 // if a is not empty, append the leading letter
                               // of w (which is capitalized)
    b +                        // append the other letters of the ref. substring
    [c]                        // append the padding string
                               // (or an empty string if undefined)
)                              //

Charcoal, 99 74 bytes

ＷＳ⊞υι≔Συη≔⟦ω⟧θＦθ¿⬤υ№ικＦ‹ＬιＬη≔ιηＦΦυ¬№ικＦＬκ«Ｆ⁼…κλ✂ι⁻Ｌιλ⊞θ⁺ι✂κλ»⭆η⎇⊙υ⁼κ⌕ηλ↥ιι

Try it online! Link is to verbose version of code. Takes input as a list of newline-terminated lower case strings. Explanation:

ＷＳ⊞υι

Read in the inupt list.

≔Συη

Assume the worst case that the solution is the concatenation of the input.

≔⟦ω⟧θＦθ

Start a breadth-first search for a solution from the empty string.

¿⬤υ№ικ

If the current string contains all of the input words, then...

Ｆ‹ＬιＬη≔ιη

... if it is the shortest so far then make it the current solution.

ＦΦυ¬№ικ

Otherwise, loop through the words that aren't contained by the current string.

ＦＬκ«

Loop through all of the potential overlap positions of the word with the end of the string.

Ｆ⁼…κλ✂ι⁻Ｌιλ⊞θ⁺ι✂κλ

If the word overlaps the string then add the result to the search list.

»⭆η⎇⊙υ⁼κ⌕ηλ↥ιι

Output the shortest string, but capitalise every letter that is the first occurrence of any word on the list.

Edit: Saved 20 bytes by only considering overlaps at the end of the string and a further 5 bytes by calculating the overlap more efficiently.

jq, 194 bytes

def p:.[]as$x|[$x]+(.-[$x]|p//[]);[p|reduce.[]as$x("";[.+$x[range(length;-1;-1):]|select(ascii_downcase|contains($x))][0]|sub("(?<m>\($x))";($x[:1]|ascii_upcase)+.m[1:];"i"))]|sort_by(length)[0]

Try it online!

Ungolfed:

def permutations:
    .[] as $first | [$first] + (. - [$first] | permutations // []);

[ permutations | reduce .[] as $word (        # for each permutation, fold:
    "";                                       # starting with empty string
    [
    . + $word[range(length;-1;-1):]           # append "", "d", "rd", "ord", "word"
    | select(ascii_downcase|contains($word))  # filter to when the full word appears
    ][0]                                      # get first (i.e. shortest)
    | sub(
        "(?<m>\($word))";                     # capture word as .m
        ($word[:1] | ascii_upcase) + .m[1:];  # uppercase first letter
        "i"                                   # match case insensitively
    )
)] | sort_by(length)[0]                       # shortest final string

The inner [ .. ][0] seems like it should be able to be removed, but it can't. This is because reduce iterates with the last object generated, rather than every one.

05AB1E, 19 bytes

œJJæé.ΔIåP}ÐIk©èu®ǝ

Based on @JonathanAllan's Jelly answer, but even slower. 😅

Input-list of words in full lowercase.

(Don't) try it online.

A slightly faster approach (replacing œJJæé with S©ε®N>.Æ}€`J) that is able to output most individual test cases (although it's still pretty slow, and still won't be able to output the first two test cases..):

S©ε®N>.Æ}€`J.ΔIåP}ÐIk©èu®ǝ

Try it online.

Explanation:

œ                # Get all permutations of the (implicit) input-list
 J               # Join each inner list of words together
  J              # Then join everything together to one large string
   æ             # Get the powerset of this
    é            # Sort it by length (shortest to longest)
     .Δ          # Pop and find the first/shortest that's truthy for:
        å        #  It contains as substrings
         P       #  all
       I         #  the input-words
      }Ð         # After we've got our shortest string: triplicate it
        I        # Push the input-list of words
         k       # Get the index of each word in the top copy
          ©      # Store those indices in variable `®` (without popping)
           è     # Pop another copy, and get the characters at those indices
            u    # Uppercase each character
             ®ǝ  # Insert them back into the string at indices `®`
                 # (after which the result is output implicitly)

Jelly, 18 bytes

FṗⱮL$ṚẎẇ@ⱮÞ⁸ṪŒuwⱮ¦

A (very inefficient!) monadic Link that accepts a non-empty list of non-empty lists of lowercase characters and yields a list of characters.

Try it online! (Although there's not much point trying more than seven total characters!)

How?

FṗⱮL$ṚẎẇ@ⱮÞ⁸ṪŒuwⱮ¦ - Link: list of lists of lowercase characters, Words
F                  - flatten {Words} -> Chars
    $              - last two links as a monad - f(Chars):
   L               -   length
  Ɱ                -   map with:
 ṗ                 -     Cartesian power
                     -> [All length N strings using Chars for N in [1..length(Chars)]]
                        (duplicates abound with duplicated characters in Chars)
     Ṛ             - reverse
      Ẏ            - tighten
          Þ        - (stable) sort (these Strings) by:
         Ɱ ⁸       -   map across Words with:
       ẇ@          -     sublist {Word} exists in {String}?
            Ṫ      - tail
                     -> a shortest string containing all Words
                 ¦ - sparse application...
                Ɱ  - ...to indices: map with:
               w   -                  {String} first index of sublist {Word}
             Œu    - ...apply: convert to uppercase

Python3, 383 bytes

S=str.capitalize
L=str.lower
def f(w):
 q,r=[(S(i),w-{i})for i in w],[]
 for a,b in q:
  if not b:r+=[a];continue
  q+=[(a[:(O:=-max(i for i in range(len(j)+1)if L(a[(-i or len(a)):])==j[:i])or len(a))]+''.join(A if u==0 or u>=len(a[O:])else a[O:][u]for u,A in enumerate(S(j)))if(J:=L(j))not in L(a)else a[:(I:=L(a).index(J))]+S(a[I])+a[I+1:],b-{j})for j in b]
 return min(r,key=len)

Try it online!