I'd check out Alibow. Incredibly affordable, yet high quality, entry-level fiberglass horsebows. You can get one for under $100 and he responds to emails very promptly for being overseas. This is what I switched to from the opposite-hand recurve, and it is night and day. If you're going to spend any significant amount of time working your way into the thumb draw, I'd recommend just pulling the trigger on one. A recurve will never achieve the feeling of a horsebow. The difference in construction and energy transfer is vast. Having an appropriate bow to test techniques on will open a whole new world of possibilities that you can then translate back to Mediterranean.
For me the most complicated aspect of learning thumb draw was deciding on an anchor. There are dozens of variations and they all modify arrow elevation. The "cool factor" of a floating anchor was something I was really drawn to, but quickly discovered very few Asiatic-style archers actually use it. It needlessly complicates the angle of the shaft and allows for inconsistencies. The exception is Chinese style where I'm drawing past my ear - there my anchor is the fletchings on my cheek. I'd recommend just drawing with your thumb to whatever feels natural and go from there. I use back jaw because arrow stays at a similar elevation to my western bows. Resting lower thumb knuckle to face, or the flat of the thumb between the joints works well for me. Many Asiatic archers also use a top knuckle in the corner of the mouth anchor, contrary to popular belief. Really anything will work - all about comfort.
One thing I've recently worked out that was very hard to find info on was finger positioning upon release. Armin really helped me with this. I was nicking my finger pretty good and what he suggested was releasing in a way that leaves your thumb and index in an "L" shape with your thumb facing the target. This way your index is always "pulling away" from the string, instead of crossing it. Also, make sure to leave the "hole" you create with your thumb and index when drawing spaced out, so it's not cramped and allows for plenty of pressure on the arrow shaft.
I shoot instinctive and hunt split finger with a recurve. My biggest concern was that learning this new style would ruin my brain and interfere with my Mediterranean shooting, but it's amazing how the mind is able to compartmentalize. I can shift between the two - different anchors, elevation of arrow, etc. - without a blip. So don't let that concern you. Initially, it will take your brain time to adjust to the arrow being on the opposite side. A good way to ease into this is to somewhat purposefully bisect your vision with the bow. Eventually you'll be able to forget about it.
As far as the "pushing," it is a mistake to ever try to add movement purposefully, I think. It's all an equal exchange of energy from tension, not a linear forward movement like the word suggests. As far as your mention of short bows, I have noticed that khatra seems far more pronounced with very short horsebows. With my long Manchu bow, trying to shift it downward with momentum seems very awkward. I usually end up naturally going into side khatra. So maybe shorter bows just naturally provide better follow through due to lower dispersion.
I know that for me, exploring these new styles has enriched the sport to a degree I could never have imagined. There's so much history wrapped up in the different philosophies and techniques, and they almost always teach you something you can apply somewhere else.
Happy to provide more observations on specific form questions, at least when it comes to what has or hasn't worked for me.