You will need to be very specific with Europcar about what vehicles you'll be allowed to take into Czech. I doubt very much that you will be allowed to take a minivan (or any vehicle able to fit all of you) into Czech, based on the Conditions of Rental (for Germany), part of which I have reproduced here for you:
3) WHERE CAN I DRIVE THE VEHICLE (CONTRACTUAL TERRITORY)
The renter and the diver are not allowed to use the vehicle outside the contractual territory. The contractual territory is Europe with the exception of the below mentioned countries which cannot be used depending on the vehicle category.
Not allowed countries for all vehicle categories:
Albania, the Baltic Republics, Bulgaria, Greece, Iceland, Romania, Turkey, the successor states of Yugoslavia (except Slovenia and Croatia), Russian Federation, Malta and Cyprus.
Not allowed countries for vehicle category full size and higher categories:
Great Britain, Ireland, Italy, Croatia, Poland, Slovakia, Slovenia, The Czech Republic and Hungary .
Should you have questions please contact our Customer Relation Serviceunder +0049 40 52018 8000.
Please be aware that You must comply with the law, in particular the traffic road regulations and toll payment obligations in the country where You drive the Vehicle. You as renter and driver are liable for all claims resulting from vehicles owner liability during the rental time.
You may find a rental agency that allows this, or revisit your issues with trains.