A VFD conversion does not use the electrical cabinet contactors to switch the motor wires, the motor is directly connected to the VFD. See the "basic 1340GT VFD conversion" thread for details of how this is done, also QMT has some info they provide. The motor space is very limited in the 1236/1340GT models there are only a few TENV motors with a 145 frame size that will fit, a TEFC will not fit. The stock 2 Hp motor is quite decent and provides plenty of usable speed range, typically 20-100 Hz, I also recommend flipping the motor pulley and running it from the large motor pulley to the large headstock pulley (requires a longer belt) and running the belt as a single belt speed. Since you have a 6 speed headstock, you do not need extend the motor speed range. If one wanted to replace the stock motor then you would need to replace the motor pulley anyway with a 2.4-2.6" single pulley (stock is 3/4" and most 145 frame motors are 7/8"). You would run the inverter/vector motor from 20-125 Hz. Worthwhile if you have a single phase motor, not so cost effective if you have the stock 2 Hp 3 phase motor. You gain a bit more usable speed range and the speed stability running the VFD in sensorless vector mode you go from +/-5 RPM to something like +/-1 RPM speed variation. So the stock motor is pretty decent and works well paired with a decent quality VFD.
The alternate motors has been previously outlined, you can do a search example:
I figured out how to get into Super Macro mode, these aren't very good but they're closer:)
www.hobby-machinist.com
"If going to a 3 phase motor, there are a very limited number of 2Hp 3 phase motors that will fit, and they all require some redrilling of the mounting holes motor/mount to fit and a different pulley (or bore the stock one to 7/8"). These motors are all TENV, these are ones that others have used Marathon E467, Y551, Y526. The Y551 and Y526 have a small encoder shaft on the back of the motor that needs to be cut off." In addition the Baldor IDNM3587T if you can find one inexpensively, the back encoder shaft is cut off with a hacksaw, you may also need to do some trimming of the belt cover. All these motor conversions were done on the 1340GT, not sure if the 2 Hp will fit on the 1236GT, as you can see it is a tight fit.
I would use the stock 3 phase motor first and then see if you want to switch it out, at the end of the day individuals tend to overthink all the what if's before they actually start using the lathe. I would use the Hitachi WJ200-015SF VFD which is 2 Hp for all these motors, it works well and there is a lot of experience using the it.
With the 1340GT you get better a much better chuck, with the PEP package, so you are looking at a 1K difference if you take out the chuck. The 1236 comes with an "economy chuck", you get what you pay for, it would be the first thing that would become a doorstop for me. Start adding a motor, your time, etc. and the difference becomes smaller. Then, start adding up all the tooling/chucks, which would be the same for 1236/1340GT and it all adds up quickly, the overall cost difference becomes smaller.