'mě' and 'mne' are pronounced exactly the same, correct? Shouldn't both be acceptable here?
"Mne" a "mě" should indeed be both accepted, but it is not true that the pronounciation is the same. "Mě" is pronounced with soft "e", but "mne" is not.
The only difference here is that "mne" sounds a little archaic and is almost not used in spoken Czech anymore.
But also don't rely on the theory "sounds the same = can be used as well". "Mě" and "mně" sound the same but are used in different cases so it actually never happens that both of them are correct, always just one. But even Czech people struggle with that and a lot of the times just don't know which is correct, so don't worry about that :-).
Does anyone know why "She probably doesn't remember me now," isn't accepted?