It would be nice if pvpers could show off our highest rating achieved on our trainer card. Maybe even a badge for tournament wins or top 3 ladder placements. The friendly battles won/lost/disconnects stats are meaningless on the other hand and can be removed.
The ideal pvp match window for me would first let you select a ladder (ou, uu, random, etc), then from there you can search for a match, view current ladder placements, maybe even some other data like the top 25 usage.
To expand it even more, if there were some extra features like a teambuilder to build and store teams, integrated damage calculator/team weakness chart, etc could take what we usually have to do outside of pro right into the game. It would definitely be a lot of extra work, but if I had to imagine the perfect game would have all of these little features. Even putting in a "notes" section would be a good start, we could write out our own speed tier charts, a team list, ev spreads, etc