The QR lounge is nice but do keep in mind that it goes into reduced catering mode from about 8pm until 11:30pm, the bar is still excellent but if you want to eat, head to BA or Qantas.

I would do a Singapore arrival card. It’s free and will reduce the likelihood of tricky questions when you board the original flight. While your timing should be more than sufficient for what you’re trying to do. Note that even with checked baggage, you shouldn’t have to clear immigration (although given SIN is gate security in T1, the amount of time it would add is minimal). Take your bag receipts to a transit desk, show them your onward boarding cards and they’ll retrieve the bag from the carousel for you and transfer it to the BA flight.