On a multi-router network, should a router (R2) behind the internet gateway (R1) run source routing
No. Source routing creates security problems and can only be used in fringe cases. Generally, it should be considered obsolete.
or masquerading if the gateway router is masquerading?
No. You only masquerade (NAT) where necessary, between different address domains (usually public vs. private). Multiple NAT stages can become very difficult to handle.
Routers within private address space or within public address space should always use plain routing.
I can ssh from a host on the gateway subnet (R1) to a server on the other subnet (R2), but attempts from the Internet time out.
You might have multiple NAT stages and have missed setting up destination NAT aka reverse NAT aka port forwarding on the outer one. Alternatively, a firewall rule is missing or your ISP is blocking the connection. Run packet captures on the gateways to debug.