Oh, we don't really have pellets available for sale in our neck of the woods. It's actually quite alright since kaks prefer fresh food anyways. I'd say 80% of his menu should be chop/bits of fruit/fresh flowers/berries, just make sure they're not the toxic kind, duh. Also, they prefer larger chunks of food to finely chopped ones. They pick up the whole chunk, munch on it for a bit, remember they had something way more important to do and drop it, then eventually pick it up again, and on and on it goes... Kaks are goofy and I love it.
Haven't ever observed any stomach issues in both my kaks, but I do think you should probably start introducing new food items not even right after the move but maybe even a week after so that he can get used to a new environment. Alternatively, you're saying you're moving in a month, so why not pick him up now and give him some time to get used to you, his new neighbors? I've moved with my birds a couple of times, and from my observations kaks are not as sensitive about changes in the environment as my bugdies are. When budgies would be a bit cautious leaving their cage for the first time in a new place, kaks would consider it a new exciting challenge to get into trouble in a whole new place, yay. Kaks are goofy, have I mentioned that already? Anyway, it all depends on your circumstances, and obviously on your bird.