There are luggage storage places in Bangkok. Also, if you work it right, you can stay at the same hotel near the beginning and end of your trip. They usually will store it for you. (Unless you are roaming Thailand for months...)
Thoughts? Sounds like fun when you say it, but when I start thinking about it, BKK to the border is probably at least 20 hours. Then you would HAVE to get off and do immigration stuff, then get on another train for 5 plus hours. Some people may enjoy that, but many wouldn't.
What I would do, as suggested by someone above, would be to catch several trains and overnight in a few Thailand towns/cities, etc. Then do the last slog in one go from the border to KUL.
When I was testing it out last week, when I selected UK and UK (passport, and current permenant residence), it listed the Embassy in london. When I did UK and USA, it listed several in the big cities (Chicago, etc). When I listed places like Malaysia, British OTs, India as tests, it would not let me proceed. I took that as there was an element of "going in for an interview" or something.
I just know that even though I have a UK passport, I will not be in London OR the USA anytime soon.
BTW, the online process is still not fully online. You can apply online, but you still need to go to the London Embassy/consulate in person with your passport. (As of about a week ago.)