
All about the temperature
It sounds like most well-informed commenters are agreeing that the problem is the fuel temperature in the fuel contacting the igniter. It's hard to imagine the problem fundamentally being pressure. If the rocket could only burn at 1 bar then it's not going to do well once the blast plug has popped off and the thrust has settled down to ~2 lb, since that will only provide a pressure back on the fuel of ~30 mbar. If it's going to work at all, it has to be self-sustaining in a vacuum.
So the solutions come down to either:
1. Use a blast plug, to contain the flame and help it to heat the bulk fuel to the autoignition temperature.
2. Heat the fuel externally, e.g. with thermite paste.
My preference is for 2, since it won't waste energy popping off the blast cap, so it should give a higher final altitude, and it seems like there's less that can go wrong with it.