This is more of a search for a solution if it exists, if not then it becomes a suggestion I guess.

Does anyone know of a way to send a friend request to someone you have had a battle, or even multiple battles with?

I've had some pretty close battles as I'm sure many have but I have no way of contacting these people to even say "Good game".

One option to add someone into your friends list is via rankings list.
Just click on the username to see his/hers replays, picture of his/hers mech and "add friend" button below his/hers mech picture.

Another option is via forums.
You just could ask their player ID or give out your own player ID.
Some users here have their forum name same as it is in game (e.g. me), but some prefer to use different name.

Not sure if you can add someone as friend during a battle though.

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Aeacus