Not "illegal" technically, but if they don't accept a copy and a cash deposit, find a different rental agency as you need to have your passport on you at all times.
Or, go to an immigration office and speak to officers... ? This group gives a ton of good FREE advice.. no like, ask agent or, go to immigration or get tons of incorrect answers elsewhere in the troller groups.. have a good day.. bye bye!
The return is the bigger issue, best for doggo, is have either a trusted family member or friend go puppysit at your house. The travel is quite hectic for the animals for short periods.. best of luck!