I am playing on xbox and was offered to work on a collaboration map.

My club-mates sent me a *.map.gbx file. I am attempting to upload it to TME so I can add it to my account and access it on xbox.

When I try to upload the file, I get the following error:
"You must compute lightmaps before uploading this map."

Is there any work around to this, or way I can work on this map from console without one of my club-mates publishing it?

The map will need to be validated and lightmaps computed no matter what you do. But your friends can upload it to a map upload activity in-game in a club, and then you should be able to edit the map from there.