Thanks once again for all the tips. In the end we had a lot of time at the airport so we went for it.
It went pretty much as stifle mentioned with a few small differences. The way from T5 to T3 took about 40 minutes and the way back about 30 minutes. However, as you need to go through the security again, I would recommend more time.

Edited guidelines:

1. Go to T5
2. Clear departures security at T5 normally
3. Go on the train to T5C
4. Stay on the train when it says get off, and hope that there are no staff there who shoo you off (they did but were OK when we told them we just need to get back to T5A)
5. Go back on the train to T5A
6. You are now in T5A arrivals/connections
7. Take the connection bus to T3
8. Go through boarding ticket check before security - lady was OK when we informed her about going to the Amex lounge, but this can be the most risky part
9. Clear transfer security at T3
10. Do whatever business you have at T3
11. Convince T3 staff to let you back to T3 connections - no need, you can just go back the way you came and get to the bus without any checks
12. Take the bus back to T5
13. Go to a BA desk and explain why you need to get your boarding pass reset to go through again - no need, the lady at the control at the bus station just scanned the boarding pass and asked what happened (simple "We went to lounge at T5" was fine)
14. Go through T5 Ready to Fly checks - not sure what you mean by this, we just went from the bus directly to the security check
15. Clear transfer/departures security at T5
16. Go get your flight.

Was it nice experience? Definitely!
Would I do it again? Probably not as it is painful process to go through security that many times.