Yes you can, you can access the key port above the tire to lower by taking off the license plate, at least you can on a 1997 and 2003 tacoma Also works on Tundras.The thing about tailgate on, is one I also believe it makes it hard to enter, and in case of a Toyota Tacoma you can’t lower the spare tire, if the tailgate is down.
I’m curious how you have the camper anchored to the truck with it shifted that far back.I have started to roll with mine on and down as well. Allows me to keep the truck box for storage and a place to stack firewood. Plus the tailgate is a giant pain to remove now days. Takes me longer to plug/unplug and fish the 3 sets of connectors in/out than to remove the gate, so I just leave it on. Probably not a option for some trucks but mine is fine. I just use the built in tailgate step and a folding stool to get in and out. Seems to work fine for me.