Can my Penguin 200 and tank handle the bio load after moving?
The Penguin 200 should be adequate for your 38-gallon, and I don’t think you need to wait to add all the fish. Your filter should be established, so you if you use the old filter with the old filter media, you won’t need a break-in (“cycling”) period for the tank. You didn’t really need to move all that water, either. The beneficial bacteria that accomplish biological filtration are mostly in the filter and on the gravel and other objects in the tank, so if you transfer these, it’s fine to use new water (with a dechlorinating water conditioner, of course).