Although the Godox website (
GODOX PHOTO EQUIPMENT CO.,LTD TT350P) only lists the 7 cameras you refer to, I can confirm that it works fine on my K-3, as well as my K-S2.
I received mine yesterday and I've been testing it today. I've tried both manual and TTL modes, with and without HSS, and it's all good. Yet to try it off-camera (I got the XProP as well), but I'm already delighted with it. The ability to do HSS not only in PTTL mode but also in manual mode is a bonus - even my Metz 58 AF1 can't do that.
Godox also lists only the same 7 cameras as being compatible with the XProP, but PF members have tested the XProP with a range of Pentax DSLRs apart from those 7 listed and it has worked with every model tested so far, including the K10D and K20D:
Xpro-P TTL Wireless Flash Trigger for Pentax - Page 2 - PentaxForums.com
It looks like Godox have only confirmed compatibility of the TT350P and XProP with the 7 cameras listed because they are the only ones they have tested themselves.
I hope someone else can give a definitive answer to your question, but it would be surprising if the TT350P didn't work on the K20D or K200D.
Incidentally, if you do decide to get the TT350P, consider getting the XProP at the same time - it's cheap in a bundle with the flash, and promises to be a good combination for off-camera use.
I'm really hoping that Godox will release a version of their higher-powered models that also does on-camera PTTL - I could see a PTTL version of the TT685 it replacing my Metz. They did the same kind of roll-out over a period of time for Sony and Fuji so there's a good chance they will do the same for Pentax.