Reverse engineering and messing with a TENS unit is totally unnecessary. The data published by the UMich team in their peer-reviewed papers and video they released, provides all the necessary information you would need to create such a device. You are quite correct that materials cost would be in the $100-ish range, but you'd probably want an oscilloscope setup to validate your timings.
I think a few people were trying to do this; one even posted a bunch of pics and some half-baked Arduino source code before completely disappearing forever as far as I know.
The tongue thing is, IMO and based on my experience, unnecessary; I would stick with the same kind of electrode to the TM area that UMich used. My guess is that Lenire is using the tongue thing because it can simply be rinsed and reused, unlike electrodes that need to be placed.
But, I think it's an insane waste of time to even be thinking about this when we have multiple products this close tot he market. If this stuff works, we're gonna find out soon, and it's gonna be commercially available. If that doesn't happen then it means it does not actually work.
The UMich Phase-I prototype devices were literally just small aluminum boxes which contained an Arduino unit, a battery, and then had an audio out port and a connection for the electrodes. Timing and intensity of electrical impulses controlled on board, this was setup through a PC as a one time deal at the beginning of the study based on my profile.