Do accusative ‘me’ and ‘thee’ have final /m/ (or evidence of a lost or altered /m/), by analogy with (non-neuter) nouns, in any Indo-European language?
"me" and "thee" never had a final "m". The Proto Indoeuropean accusative forms were simply "hme" and "te". (Different reconstructions see here)
In Latin, the vowel was lengthened: mē, tē. Indoiranian added an "m". Proto Germanic added a "k" in the accusative case, a "z" in the dative case.
The source of the final "k" is either from the particle "ge" which still existed in Ancient Greek, or comes from the nominative "heg" (Proto-Germanic ik, German ich, Enlish I).
The modern English forms "me/thee", however, might rather derive from the dative forms "miz/þiz" which the odd ending "z". The Proto Indoeuropean Dative was "tebhi/toi" or something like this. The Germanic formation is unique here.