Python, 157 bytes, average byte value 79.2774

always produces a string with the lowest possible average byte value

lambda l:''.join(chr(97+[x for(x,_)in sorted([(c,-len(list(n)))for(c,n)in groupby(sorted(l))],key=lambda t:t[1])].index(c))for c in l)
from itertools import*

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