Until her much regretted demise age 17 our cat caravanned with us everywhere.
Suggestions.
Cat bed at home in a medium size pet carrier, but withfront grille removed. In transit, carrier, with grille, on backseats of car but facing foward not across.
In caravan bed in carrier without grille. Ideally in van with seperate toilet room and shower. i made false floor for shower which held fod bowl and litter tray. Propped shower cubicle door slightly open so cat had access any time.
Walked on lead but with small dog harness, not just collar as cat's necks are relatively weak. If on a nice shady pitch put out a stake and long lead but always with length to get back into van if scared.
When getting very old put small heating pad under bed, so it could be warm - sort of thing sold in chemists for easing pain in (human) joints.
When at home and van door open, cat always went into van for siesta.
Great conversation starter when walking her round pitch on lead.