Hello sorry I've only just got the notifications through, you can hold more than 1 passport I know the UK try to be funny about it but there's no legal law that you can't I can co firm how easy or hard this would be if you like? You do not need to be native English but you do need a degree to get into one and get paid enough however you can get a TEFL and work in any Thai school or do online teaching there's options out there don't listen to the negative comments I have for so many years and stopped me achieving my goals but now I'm just focusing on what I can do and listen to people who actually know what they are doing and have done it xx
If the children are half Thai from my understanding from someone I know who is half Thai you can get them a Thai Passport & still keep your other passport for them too which would help alot for them.
If you can get a job in an international school then they would get free/very cheap education there and would be able to get a visa that way.
I don't want to give to much advice on it that's nit necessarily the answers you need but the children going through a visa with their father assuming he has legal stay in the country will be the best and easiest route for them & yourself x