Seems like any trade functionality except sending a trade is not available via gamepad UI. When sending a trade, the trade window will not appear after the second player accepted it. See screenshot for evidence.
Getting a tade request wont even show a popup to accept. The players have to move away from each other to cancel the trade interaction.

How to reproduce:
Just change to gamepad UI and get traded to see no popup. Move away from each other to cancel.
Trade another player by holding LT or something like that when next to a player. The other player can then accept and they will see a trade window. The gamepad UI wont show it.