Does it scan the first x minutes when I join causing lag?

No. The Anticheat does not cause you "join lag". The AntiCheat always scans you. That means no matter whether you check-in 20 minutes or in 5, it is always doing the same thing. Therefore no, your server "lagging" in the first 1 minute or whatever is not related to the AC because it is always doing the same stuff, no matter whether you are checking in 5 minutes or in 20 minutes.