Instead of it simply being logged in. Tie it to Challenges. When challenges are completed on a server, they give exp plus bonuses to Battlepass exp. But you get unmodified exp to your decay gauge. Which as the OP suggested, decays over time.

The gauge can be adjusted in server settings to decay at faster or slower rates and have a higher or lower max value. Giving server owners max control.

The Challenges are some of the best metrics of activity and this would give them a use when we complete a battlepass.