The router IP doesn't really matter, nor does the subnet mask associated with it, but the robot network must be the private network 10.xx.yy.zz. You have no choice there.
I think that's a common misconception about routers. A router uses it's routing table (IPs and associated subnet masks for all connected devices) to figure out what to do with message packets, not it's own IP/mask. It's own IP/mask only matter when you are trying to talk directly to it, as when you want to login to look at and change the router settings. If we were restricted to having an IP address in common with a router, then no one would ever be able to talk to another computer out on the Internet.
I'm not clear on the concept you are going for.
If you really used the school's router as a "bridge" for the robot, then it would have to be mounted on the robot...
Do you just want to be able to access the Internet from your Driver Station/programming laptop?
If that's so, then why not just run two separate networks from your laptop?
One NIC set to 10. and the other to DHCP from the school's router?
Here's a network briefing that might help:
http://team358.org/files/workshops/NetworkTopology.ppt