Do you have a scope? It would be good to know whether the p-fet turns fully on when you press the button. In other words, what exactly isn't working? Is the p-fet not turning on at all, or is the p-fet turning on but the n-fet isn't?
Well, I think the problem almost has to be the different threshold voltages versus the original design mosfets. It's possible that playing with the capacitor and/or resistor values could get it to work at lower voltages, but in the end you may need to use discrete mosfets with the right threshold voltages, if you can find them. Still, it seems like there ought to be a way to make it work with 3.7V input.
Edit: If you start it up at 4.4V or more, then lower the input voltage, at what input voltage does it turn off?
Edit2: This may be crazy, but I wonder if replacing the n-fet with an NPN might make it work. Well, that would be a different circuit. Anyway, I just wondered.