Your entire post boils down to, "Prove my theory is not true." Think I'll pass. It's just a bunch of assumption at best.
No, it is based on the fact that they clearly raise money through channels dedicated to international aid and from donors who want to help them use their expertise for specific projects abroad. It isn't an assumption, it is just common sense.
If you want to act all reactionary and just assume that all of the 2% they spend on foreign aid would still exist and could be spent on the UK if they weren't doing international projects than that is up to you. It would be clear from everything you can read and all the evidence we have, that you would be wrong though.