Move a six-router network from RIPv2 to EIGRP without an outage

A company runs RIPv2 on five Cisco routers in a ring (R1 to R5) plus an internet firewall, FW1, that hangs off R1 and only supports RIP and static routes. Failover takes minutes and the 2 hop path over a slow backup link keeps winning against a 3 hop path over fiber. You are asked to move the routers to classic EIGRP AS 100. The business allows short night windows, and nobody wants a big-bang cutover. FW1 sends R1 the default route by RIP over Gi0/0/2 (10.0.1.0/24, FW1 is .1, R1 is .2).