I also put my pets in the neolodge. It makes me feel bad if they are starving. Even though it's not real
If you want to get rid of it, there are enough ways. Put it in shop, SDB, or donate in to the money tree. Or discard it.
My pets have strong wills! They sustain a dying status for long periods of time, longest of which was a few years. The Earth Faerie sometimes feeds them for me. When I want to battle I'd have to feed them too. I'm biased...
i feed them every day and the're always bloated, but i just give them omelet or jelly, sometimes very cheap things that i found in the igloo.