Do NOT put a DHCP server on your network in this situation. Your server and their router will start fighting over client leases and probably cause a packet storm, knocking out the network. Especially if your server starts farming out addresses on a different subnet, that is going to cause all sorts of problems for you and them.
With that out of the way; this is a really weird set up. Is there any kind of segregation between your networks? Is the router handing out addresses on two different subnets - one for you and one for them? I would be wary of putting anything company sensitive on this network.
The safest solution is to get your own router and connect the WAN interface to their router. Then connect the rest of your network to your own router. To your router, their router is just a gateway device, they have no visibility of your network, and you are in control of the subnetting and other network services on your network. You can either enable DHCP on theyour router or on a separate server.