I think this is a worthwile idea. There are so many great benefits to being a supporter like bookmarks, buddys, scoreboard, quotes, roster numbers, training history etc that it may well attract a lot of new supporters. They don't get the best feature which is the private league.
A week a season is way too much and would be a major disruption to revenue, but a one off "Christmas Present" might be something the BBs could consider, but they also have to weigh up losing one weeks supporter from everyone who already has it against how many new supporters they would get. I have no idea at all how many supporters there are but if there are 60K players there may be 3k of supporters. It costs about $1 a week to be a supporter so it would cost $3,000 for this promotion. That means they would need a lot of people to switch from non-supporter to supporter to make it even worthwile.
I guess one possibility would be to not give the current supporters a free week, only give that to non-supporters. Instead give current supporters some other non-advantageous cosmetic only benefit like the ability to rename a player or their next draft pick, to be invited into a special supporter worldwide pl style cup or perhaps being able to give an "accessory" (headband, goggles, facemask, tattoo, mohawk, pink hair etc) to one of their players.