In regards to the OP's question.
Reman Toyota would be a good option, though I think you only get the full benefit of the limited warranty (12mo/unlimited miles IIRC) if it's installed by a dealer.
I could be off-base on that, but I vaguely remember a situation with a Toyota reman'd transmission that was installed by a 3rd party shop. I think in that case we charged the customer diagnostic time for "proving" to Toyota that it indeed qualified as a true warrantable failure before they would honor the warranty (I think they were waffling a bit since they "couldn't be sure it was installed correctly if it wasn't done by a dealer or some BS).
I'll bug my parts manager about it this week and get back to you for a definitive answer; even Toyota's own statement leaves some vagueness...
UPDATE- 12mo/unlimited miles, part only warranty unless installed by dealer. At most you should only have to fill out their worksheet which will ask for measurements of stall speeds line pressures, etc. of the failed part if you do need to ever warranty it and it wasn't installedby a dealership. Hope that information helps.
Plus, all Genuine Toyota Remanufactured Products are covered by a 12-month unlimited mileage warranty that’s honored by more than 1100 authorized Toyota dealerships in the continental U.S., Hawaii and Alaska. See dealer for details on limited warranty.
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Purchase Genuine OEM Used Auto Parts for Your Toyota
But I will say that I haven't heard the best of things regarding Jasper and I want to say AMMCO in the last decade or so; have had a couple guys who owned shops or used them in both the Bay Area and out here have issues with their quality and warranties. I have never dealt with them directly, so take that how ever you want.
In terms of rebuilding the transmission? It's doable, I just don't know if you can get new "genuine" parts? I don't think Toyota would have much of anything in the way of parts for the vehicle, so you would have to do your research on aftermarket offerings in terms of seals, gaskets, new bands, clutches, etc. Also rebuilding an automatic transmission yourself is a bit of an endeavor; it's not quite like rebuilding an engine or differential. There is a lot of measurements you're going to need to take and compare to either a copy of the FSM or the transmission manual to determine what things will be replaced and if you need specialized parts (i.e. "Shim Kit B" because your total stack tolerance exceeds X and so you pick from a table of parts). It was both a fun and very technical experience when I rebuilt automatics back in school (fun memory, one of my partners managed to install the one-way clutch completely backwards, every roller and spring, etc. was impressive... even if we had to tear it back down to fix it.)
Post Note- (If you don't care about stupid technical, historical, pedantic stuff, skip the wall below, you've been warned.)
[MENTION=164664]brillo_76[/MENTION]
So having gotten sucked down a rather deep rabbit hole and digging around in the dark corners of TIS and the history of ATFs... I am going to say that the most "official" document in regards to ATF T-IV's "compatibility" with Dexron III fluid is in the attached PANT Bulletin. It is actually the only place I see any mention of an "ATF T-III" outside of an obscure SAE paper regarding approaching early OBD-II false misfires due to how the TCC worked back then, this required a redesigned TCC system that allowed some slippage and thus a new fluid was supposedly needed? Besides this PANT bulletin and that SAE paper, "ATF T-III" doesn't seem to exist in that specific name. Coincidentally a few years after this paper the Dexron-III(F) [GM Spec GM6417M] was being phased out for the newer Dexron-III(G) [Also GM Spec GM6417M] and with that came a slew of new licensed variants including ATF T-IV (which has additive packages specifically to deal with the newer way the TCC was being allowed to slip to prevent false misfires, oddly enough for the AW transmissions this was accomplished by just duty cyclin the TCC with a new linear solenoid rather than new friction material?). In '06 GM discontinued licensing for Dexron III and with the introduction of Dexron VI they basically washed their hands of the prior specifications by issuing a circular logic:
All DEXRON-III licenses expired at the end of 2006, and will not be renewed. Beyond that date, General Motors will only support the use of DEXRON-VI fluids for use in Hydra-Matic transmissions. Fluids claiming DEXRON-III type performance continue to be sold under abbreviated names such as Dex/Merc and D/M, however, since the DEXRON-III licensing system no longer exists, these fluids are not regulated by GM in any way.
There is also an unsubstantiated (well, I say unsubstantiated because I can't find a picture of it) claim that a Volvo guy in Europe bought some Genuine Toyota ATF T-IV and it said on the back: "Meets performance requirements of GM6417M (Dexron-III G)" Additionally Toyota seems to have played fast and loose with ATF specifications outside of the US (particularly in regards to the Camry based on what I was seeing pop up being referenced commonly), which doesn't surprise me. It also seems like there may have been a reformulation of ATF T-IV by Exxon-Mobil (mainly because their own JWS-3309 was updated to blanket all prior Dexron and Toyota "T" fluids).
The amount of conflicting information from Toyota itself over the years and supporting (though somewhat circumstantial if we're being honest) evidence seems to indicate that Toyota "T" fluid was just their way of trying to play with licensing around Dexron, eventually GM just abandoned licensing it and it seems like it was just passed along to substitute ATF T-IV for Dexron applications (though from chatting with older guys, parts manager, etc. it seems like this was done more informally than one would expect.) Far as I can tell from the actual fluid specifications to following the history of the ATF fluids Toyota used and knowing that fluid specs are usually set by the oil industry not the engineers making the transmissions; T-IV is just Dexron III(G), which superseded the older Dexron III(F) fluid. The supersession was never officially acknowledged because of licensing agreements and by the time those agreements expired, it didn't really matter because that specification because ridiculously generic (i.e. all the "Universal ATFs" that basically blanket all the Dexron & ATF T specifications among other Dexron variants).
I don't think this is really the "end all be all," of this debate... but it was really annoying me that I couldn't find any specific information on T-IV or T-III and that basically all Toyota/Lexus Parts/Service switched to T-IV being a replacement for prior Dexron specifications but I couldn't trace a paper trail or documentation for why. So like I said, got sucked down a deep rabbit hole trying to remember a lot of old things and figure out the context of so much conflicting information from the manufacturers.
At the end of the day, using what the factory specifies won't hurt anything; but I think at the time of this writing that ATF T-IV (either Toyota or aftermarket) spec fluids should be formulated to encompass prior Toyota ATF callouts for Dexron III or Dexron II. I'm going to go to sleep now though, I am tired.
It's extremely rare but this particular transmission may slip if you use T4. 
