Dude, get the 1/4 inch strip. It will close the gap to a tight fit. I had the same problem with my very first cage. Of course now I know better and build my cages scapeless...did I just make up a new word?
I don't think the mounting tape will work. It's got adhesive on both sides (that's what double sided means) and while you could leave the protective paper on, it wouldn't hold up to more than one close of the door.
Weather stripping is probably the best answer. Given the size of the gap, felt might be a better option than vinyl. But I think either one is going to be the ticket.
I know what double sided means, thats why i said,"It's double sided, but i'm sure you could just use one side if you want". Why would it not hold up to one close of the door? I think felt my harbor bacteria if it gets wet too often, but I'm no expert.
I build my own cages, and for the most part, all pieces are so tight that even tiny crickets can't escape. However, the door has the gap where the hinges are, and where the bottom removable panel meets the bottom of the door. Mine are all aluminum. What I did was use some of the left over aluminum frame pieces and just put a runner on the inside of the cages right behind the hing gap and panel gap. I would post pics but I'm at work. Post a couple of good close-ups of your cage.
u could use a pc of velcro tape just use the one softer side and stick to either the door frame or the jam, it should compress to fill smaller gaps, if its a wood cage you could put a lip on the in side that you door closes against to block them