The short answer is that solid mounting the vehicle plug might be the easiest answer to help you out.
Bracket link
I had a 79 Ford with the
round 4 pin plug mounted in place of one of the license plate lights in the bumper and I really liked it. Being solidly mounted made it easier to plug/unplug than the two hand wiggle of putting the dangling flats together. In my mind it made a more reliable/better connection. I also liked that the contacts were somewhat more protected by the flapper and up out of the spray/mess that comes with winter here.
The downfall is that it's not the standard (wish it was) any small/mid size trailer you will ever have or come across is going to be the flat so at best you convert every trailer you ever own to a round plug (not hard), may need to convert it back if you sell it (harder but not bad), and always carry an adapter just in case (at least I did anyway)
My current Tahoe has a 7 wire round solid mounted under the bumper so I have a store bought adapter that I leave with my boat trailer and one that I carry. It's on my list to add a round 4 pin and convert my trailers but for now it's not a very high priority.