I rarely use any other nozzles on my Smith Little Torch than sizes 6 and 7. The sizes below that do have a tendency to blow out.
Personally I’ve always fed in the oxygen “on top of” the propane, not the other way round.
For tips 6 and 7 you will need about setting 1 or 2 on a 5L oxygen concentrator. Give your OC about five to ten minutes after power-up before you start trying to use its oxygen.
As Bernard says, sensitive regulators are important.. Does your propane regulator have a dial? If so, what’s its max operating pressure? You will only need a few psi for the Little Torch, and not much more for brazing small to medium work with the BOC Sapphire.
What nozzles are you using when you use your BOC Sapphire? Not the swaged nozzles designed for acetylene, I hope. They’re inadvisable for use with propane.
If everything is set up right, you should be able to braze (i.e. with bronze, not just silver solder) very easily with your two torches, providing the work isn’t too big.
Presumably you didn’t get this manual with your Little Torch. It will help. Recommended pressures for tips 6 and 7 are actually 6psi and 8psi.