Just take a little tank media from you're cycled filter and throw it in your axie tank. Should be just fine. You don't need to use all the tank media, and if you have a HOB filter with just 1 cartridge, you're tropical tank would probably be ok if you switched the catridge out, bb lives on more than just the filter. Just keep a close eye on ammonia, nitrite, and nitrate.
Oh, and Elena if you keep your filter media wet while you move the tanks over breaks the filter should stay cycled. I used to have to move my tanks for college as well, and I always through my filter media in a cup of tank water, and my tanks usually stayed cycled