Asking University to reimburse renting a car
Yes, this is acceptable. 

At many universities, adding a few days stay at your conference location in a far-away country (using your own accommodation arrangements) is allowed and sometimes even encouraged. Especially when this allows you to take cheaper flights. This would be something similar, using the car because it's required anyway, and adding a few extra recreational km you'd pay for yourself. 

Renting a car might be an issue for some European universities, as they have rules to encourage public transport usage. The 1/4 share of the car is something you'll have to discuss with your administrative staff. They might want to reimburse 1 person for the whole thing, or split it over multiple projects. Just ask to clarify, they'll also be happy to know exactly how and what they're reimbursing.