there are a Facebook group Bangkok Pet Lovers. I suggest you maybe ask on there about flying with your cat in cabin from the UK. I am not sure about UK specific restrictions that would prevent you from flying with your cat in cabin but in terms of airlines I believe you have many options.
I came to Thailand with my cat in November 2023 (from South Africa) and arrived on a tourist exempt visa waiver (30 days) and had no issues / problems / questions from animal import customs when they issued the import permit for my cat, regarding my visa status or length of stay in Thailand..
I flew with my cat with me in cabin and did not use an agency. It was really a straightforward easy process to get my cat into Thailand.
I am from South Africa and I had South African police clearance but for my Non B visa they only wanted Thai police clearance. I am not sure what the rules are and if it will be the same for you. This was just my experience.
There is plenty of negativity in the comments here. I came to Thailand in November that is in the middle of school terms and found a teaching position in Bangkok within a week. There might not be that many position available as during the so called hiring window but it is not impossible.
In terms of accommodation, I found a monthly Airbnb rental which is great as it does not require you to commit with a lease, it is all inclusive and does not require a deposit. It gives you the balance between stability and flexibility.