Please disregard the below, as the bonus mentioned doesn't apply to QR flights.
One other thing to bear in mind, which may or may not impact on when you decide to do your 4 BA flights, is that you get a 25% base Avios bonus for Bronze.
Assuming you first did 4x QR flights, with say a week at your destination, your TP balance would be 140, 280 <destination> 420, 560. You’d only “hit” Silver after your third flight, but as this is a transfer in Doha and there is usually at least a few days delay in crediting, you’d only really have Silver after your long haul QR flights are over. You’d therefore get the 50% Silver bonus Avios for the 4 flights, but as these sound like they are short haul, that may not add up to much.
By contrast, if you could fit in at least some of your four flights before your QR trip, sufficient to accrue at least 20TP, your TP balance would be: 20TP, 160TP, 300TP when you arrive at your destination. If you were staying there a week, your status would have updated to Bronze, so your return long haul QR flights would earn 25% bonus Avios.
Obviously, this depends entirely on how flexible you are about those four flights. If you need to make them anyhow, and that is after your QR flights, it probably isn’t worth worrying about for the relatively small bonus. On the other hand, if you are making a short haul trip or two for “fun”/primarily to accrue the extra TP, you might have the flexibility to do these before your QR trip. It will also depend on the relative cost of flights, of course - there isn’t much point paying say £400+ for very short notice flights simply to rack up 20TP before your QR trip, if doing it 3 months later is < £100 and you’ve got that flexibility.