I had a similar issue on my 2001 sx150. When first started the motor runs fine for a minute or so. It then slows down, eventually sputtering and stalling. After that it runs fine all day.
It is not the VST, filters, plugs, thermo switches, thermo sensor or O2. All tested OK. Pumping the primer bulb when the motor slows down does not stop it from stalling and that makes me think it isn't fuel starvation. There is no excessive smoke and that makes me think that it isn't too rich a mixture either.
I pulled and tested one LP pump and it appeared to be OK. At least is want leaking on the back as folks mentioned. I didn't test the second pump since I have to unbolt the VST to get at the LP pump bolts.
So after many months of head scratching, I decided to raise the idle as the motor started to slow down. I expected that would raise the normal warmed up idle speed but it didn't. It just stopped the motor from stalling. It has been like that for a few months. I really don't understand what is going on and will do more in-depth testing in the off season but for now raising the idle 'fixed' the problem.
Let me know what you find out.