Belgium isn’t that big, I drive from Antwerp in the morning to the track and I’m still there as one of the first. The hotels near the track up their prices to ridiculous amounts, I do stay there for the WEC (B&B on walking distance from the track) but that same one is ridiculously expensive during the GP weekend.
If you want to see another part of our lovely (I lied!) country I’d say look for a place to stay closer to Brussels or Charleroi and rent a car.