Wow! How interesting. I have never seen VIRL fail like that before. It was indeed a flawed loading of my configurations that was the issue with the demonstration. Check out how smoothly our intended configuration went after reloading the VIRL topology!

What is surprising about this whole thing was that I cannot believe BGP worked at all!

R3#conf t
 Enter configuration commands, one per line.  End with CNTL/Z.
 R3(config)#router bgp 65213
 R3(config-router)#neighbor 192.168.0.5 remote-as 500
 R3(config-router)#neighbor 192.168.0.5 update-source lo0 
 R3(config-router)#neighbor 192.168.0.5 ebgp-multihop 2
 R3(config-router)#end
 R3#

R5#conf t
 Enter configuration commands, one per line.  End with CNTL/Z.
 R5(config)#router bgp 500
 R5(config-router)#neighbor 192.168.0.3 remote-as 65213
 R5(config-router)#neighbor 192.168.0.3 update-source lo0
 R5(config-router)#neighbor 192.168.0.3 ebgp-multihop 2
 R5(config-router)#end
 R5#
 *Mar 31 16:48:10.197: %SYS-5-CONFIG_I: Configured from console by console
 *Mar 31 16:48:10.843: %BGP-5-ADJCHANGE: neighbor 192.168.0.3 Up 
 R5#
 *Mar 31 16:48:11.700: %PNP-6-PNP_DISCOVERY_STOPPED: PnP Discovery stopped (Config Wizard)
 R5#
R5#show ip bgp summary 
 BGP router identifier 192.168.0.5, local AS number 500
 BGP table version is 1, main routing table version 1
 Neighbor        V           AS MsgRcvd MsgSent   TblVer  InQ OutQ Up/Down  State/PfxRcd
 192.168.0.3     4        65213       8       8        1    0    0 00:03:42        0