how to select a different virtual server's current pool?

Well, we ended up going with this solution:

iRule activated on app VIP:

when LB_SELECTED {
  set static::app_current_pool [LB::server pool]
}

Every time a pool member is selected for the a connection through the VIP, the global var 'app_current_pool' is set to the pool name.

The final iRule on the web VIP:

when CLIENT_ACCEPTED {
  HTTP::enable
}
when HTTP_REQUEST {
  if {[HTTP::header value "Upgrade"] equals "websocket"} {
    if { ( [class match [HTTP::uri] starts_with WebSocketUris ] ) } {
      if { [info exists "static::app_current_pool"] } {
        pool $static::app_current_pool
      else
        pool app_default
      }
      HTTP::disable
      return
    }
  }
}

Major change here is if the var exists, use it's value. Otherwise, run with default.

This has been tested and works.