Sup G.
Overall, kudos, because you're thinking about everything that needs to be changed to make everything work. A lot of pre-swap folks really think it's IG easy and it all just plugs in there like it was meant to be. It isn't.
But right up front, if you want a running auto 2JZ-GE VVTi, especially with an A650, use your build budget to buy an IS300 instead. Not kidding.
A 2JZ-GE VVTi engine is almost always a front sump engine, so you'll need the rear sump oil pan assembly, pickup tube, dipstick etc to even fit the engine in your engine bay with JZ swap brackets. Going rate for that oil pan set is $550-750. Figure with engine brackets you'll be around $800-1k just to physically bolt the engine into your MK3. That doesn't touch a single thing with transmission, wiring, accessories like PS and AC, gauges, anything.
Keeping the auto is adding a very big layer of expense and complication here when it comes to wiring and transmission control. You'll need to spend some very real $, likely on the order of $1500+, to keep the stock 2JZ-GE VVTi ECU and auto trans and have everything wired up to actually run and drive. The MK3's speedometer is cable driven so unless you're running a MK3 era transmission that can drive a speedometer cable (the A650 is not one of them) then you're rolling with no speedometer. A/C will take custom lines and an R134A conversion at the minimum, and hopefully that custom wiring harness does the trick to make that go. PS lines will be custom to connect to the MK3 rack but otherwise won't be too problematic.
Your stock 7M-GE fuel pump should run a 2JZ-GE just fine, but you'll need some fuel line changes to connect everything.
Clutch fan, radiator, exhaust, heater lines, drive shaft, trans mount etc will need to be sorted (as you already guessed) and gauge sending units from the 7M-GE to run your gauge cluster will need to be installed somewhere and wired... the list goes on.
The 2JZ-GE VVTi is the very cheapest JZ for a reason, in that it is an all-or-nothing build. It's ideal if you're going to install forged rods & pistons, get it set up on a good standalone, and rock a proper manual trans from the start and set it up for a big turbo build right away, like SF's own
@omicca did with absolutely fabulous levels of 'tism and attention to detail. Spoiler alert - it ran about 5x over your budget, if I remember right from his posts.
Now, hypothetically, if you got a complete wrecked IS300 or GS300 of the correct vintage and diligently moved EVERYTHING over into the MK3, changing wiring lengths and mounting brackets as needed to make it all work, you could hypothetically have what's basically a complete IS300 or GS300 inside of a MK3 shell. Historically I have seen very few folks achieve success with that methodology as the HVAC, gauge cluster, and misc plumbing always turns into a mess. The stock body electronics and CANBUS stuff from those cars will probably cause a great deal of errors and problems as the whole car's electronics will want to talk to each other, which is another reason why a standalone ECU and doing it all from scratch is a more proven and achievable path.
Overall, I think spending the money to make a stock ECU and stock auto trans run behind a 2JZ-GE VVTi is a dead end and you'll end up chasing wiring problems and error codes and when it's finally all running and driving you'll have spent thousands of dollars and hundreds of hours to have a MK3 that's probably slower than a stock IS300. Not exactly a great ROI. Plus you'll still need a full build and standalone and all kinds of stuff to finally turbocharge it as the gods intended.
If you're dead-set on an auto trans, and want a near-stock but cool setup that you can just drive and enjoy - I'd honestly work to bump up your budget to cover a nonvvti 1JZ-GTE w/auto trans, ideally out of a JZA70, and source the maintenance parts/rear sump oil pan/engine swap parts etc & get a Kaizen or similar custom engine swap harness. From there it'll be basically plug and play compared to the headaches of keeping a stock 2JZ-GE VVTi ECU happy. Plus it'll be a 1JZ-GTE already. If what you want is a 1.5JZ someday, that sets you up wonderfully to eventually drop a 3.0L shortblock in there, convert it to a manual trans, and go single turbo with a standalone and all that good stuff.
Finally, dude, if you think $250 for a clutch is expensive, look at what a quality aftermarket clutch costs for a 700+hp setup and reconsider your perspective. I've spent literally ten times that much money on a clutch. These cars are awesome, but they are no longer cheap to build. Not that they ever really were cheap to build...
Oh, to directly answer your question - if you want the 2JZ-GE VVTi to run on a stock ECU, it'll have to run the A650E as well. It'll have no idea what's going on with a 7M-GE A340 and trying to convert that to run on a JZ engine even with a standalone is just asking for trouble. You can probably find a good A340E from a 2JZ-GTE VVTi Aristo engine set for cheap, though, as those transmissions were routinely set aside for a manual during the countless 2JZ-GTE swaps that happened 'back in the day' and once you're set up on a standalone that can run a 2JZ-GE VVTi, it's fundamentally the same sensor package and engine profile as a 2JZ-GTE VVTi as far as trans control etc.
