You can try the BRG the other way hoping the agent sees sense but the way I suggested would work especially if you intend on booking 5 nights but say, nights 1+2, 4+5 or 1+5 are way more expensive so wouldn’t win in a BRG for the reasons you stated.

If the rates are similar for each of the 5 nights, I would guess that they’d compare all 5 nights for the same room type you booked for the paid nights and whether it’s flexible or non refundable. You’d need to specify clearly in the notes. In my view. If you’re able to get the BRG approved for all 5 nights if it’s a cash booking, you need to calculate whether it’s actually worth using reward points. Also factor in the 5th night free with reward points. I’d probably go for the BRG for the whole duration(all cash) and if approved, keep the points for another time.