The Vampire would not heal
The way I read this (if I take your quoted text) is to break it down into individual sentences:
Hit: 7 (1d6 + 4) piercing damage plus 10 (3d6) necrotic damage.
This is clear; the bite, if it hits, does this much damage, and of course, the necrotic damage would be halved due to the resistance to necrotic damage that the Necromancer has.
The next sentence is:
The target's hit point maximum is reduced by an amount equal to the necrotic damage taken, and the vampire regains hit points equal to that amount.
Taken in isolation, since the Necromancer cannot have their maximum hit points decreased, the amount that it has been decreased is effectively 0, regardless of the damage taken as per the previous sentence. Therefore, the vampire would heal by 0, since the vampire regains hit points equal to that amount, which is 0.
Another way to interpret this is that since you cannot have your maximum hit point reduced, this whole sentence cannot apply, so rather than the vampire regaining 0 HP, the vampire simply doesn't get to regain anything since you can't have your maximum hit point reduced. Either way, they resolve the same way; the vampire gets nothing.
This also matches the flavour of the Necromancer's class feature, Inured to Death, which reads "You have spent so much time dealing with undead and the forces that animate them that you have become inured to some of their worst effects" (PHB, pg. 119); since they would have gained a certain resilience to their life force, they cannot have it drained out of them as a normal mortal could. Of course, they can still take damage...