I'm afraid there is no pay-per-entry lounge at Terminal 5. Even if there was, I think it would be far too long a time for you and your child - it will be rather boring as you are likely to have about 6 hours free between flights.

I would advise you to get a day room at Sofitel at T5 or Yotel at T4 (T4 will require a bus or train rides but it is not far), or at one of the off-site hotels for the length of connection, assuming you have no impediment to entering the UK.

I would not plan on doing sightseeing after a TATL in WT/WT+ or if you can't sleep regardless of the class of travel, especially with a child. You (or your son) may well find yourself too tired to do anything and regret not having booked a day room.

I travel a lot but after a few of my recent "bad" flights where I was kept awake by noise in J/F I couldn't even face sitting in the lounge and called up hotels to find myself a day room despite not having planned to do so in advance, twice in the same trip. I felt a bit foolish that I didn't book them in advance as it ended up causing me extra work and costs.

Day rooms are usually cancellable on the day, so find out what's on offer (make sure you check the cancellation conditions), and if you happen to be happy to do some sight-seeing, cancel it and go sight-seeing. That way, you can choose on the day. Besides you'll probably want to have a shower etc, which a day room will be rather useful for.

It's just really nice to be able to have a shower, stretch out on the bed and do random things when you are on a transit after a long-ish flight, especially in WT/WT+.