Take him with in his small cage to travel there but move him back to his regular cage at Squeak Home Squeak.
Could you bring something more like a 40cm by 60cm cage?
This Pets At Home might be alright as temporary accommodation. Jasper has one for long travel trips (he is a dwarf so I can fit way more in). You can manage an 8-9inch wheel in there, but if you want it on the bars, it won't be easy. I ended up binning the shelf and wheel as they are just so impractical.
Substrate depth is ok.