An Infiniti Q45 reman'ed transmission from Joe is about $1750 to $1800 (after core return). Including labor (and installation of a transmission cooler,) I had mine replaced at a local Infiniti dealer for $2750.
Sorry to hear you have been having problems. Ouch! These guys are definitely clueless if they think that the transmission is not an Infiniti - others do not fit. Even the Nissan 300ZX auto transmission (the closest possible) would have required some modifications that, in all likelihood, was not done. So you probably have an Infiniti Q45 transmission in there, but it sure looks like they did not give you a good one, or it was installed incorrectly.
Did they clean out the radiator (where the ATF fluid flows) when they installed it? Better question: Did they even hook it up properly to the radiator?

If you do not have an external cooler, the ATF must flow through the radiator to be at the proper temperature!
Z