I want to create a Abjuration wizard (PHB, p. 115-116) that uses the Life Transference spell (XGtE, p. 160) and the Arcane Ward feature to heal people, but I've heard people saying Life Transference is bad due to how it interacts with resistance (half the damage taken = half the healing).

That is pretty relevant for me, because of the Spell Resistance feature that Abjuration wizards get at 14th level.

What if Life Transference worked like the Overchannel feature from the School of Evocation wizard (PHB, p. 118): necrotic damage that ignores resistance and immunity?

I believe that would be very thematically appropriate considering the fact that it is a self-sacrifice spell. I know that this is all up to the DM, but I want to know: Would that be unbalanced? Could that somehow be exploited?

This is perfectly reasonable

Of course, it'll come down to what your DM says, since this is not Rules-as-Written. But Life Transference already has a quite substantial opportunity cost associated with it, and all this change does is revert to the mean, so to speak: instead of the feature being weaker for Abjuration Wizards, it's instead normal strength.

So if I were DM, I'd be perfectly fine allowing this. The few possible exploits are things that would be exploitable for non-abjuration wizards anyways, so I can't even think of what might constitute an exploit particular to this combo.

This would not be imbalanced.

The features like Spell Resistance and Inured to Undeath (both for wizards) are perfectly fine on all other characters, and life transference works as intended on most other characters.

The question, therefore, is "does having both Spell Resistance and life transference work as intended on a character with both make it overpowered", and I think the simple answer is no, because the combination of the two doesn't make the spell or the feature any more powerful, only removes the detriment to bring it back to level with all other characters.