With IBM's assistance, we have determined that the LPARs lacking the TCPIP config parm 'IPCONFIG SYSPLEXROUTING' will not work. Without it, VIPADISTRIBUTE was overriding the DESTMETHOD of BASEWLM with ROUNDROBIN instead. And with the client starting up two IP connections that need to hit the same LPAR, at the same time, ROUNDROBIN will not work. Therefore adding 'IPCONFIG SYSPLEXROUTING' to these LPARS has been successful.
If you still see issues after that, there is another option to use with VIPADISTRIBUTE and that is the one called TIMEDAFFINITY. The default, when you do not code it, is 0 (zero). So if you have to, add the option and set it to a value of 2 (seconds).