I suspect that the burner control is nonlinear, meaning that turning one click at the low end of the range doesn't increase the flow by the same amount turning one click at the high end of the range.

Most valves will be linear only for a given range at a given pressure.

I think your strategy should be to control with the burner valve, not the regulator, BUT you should play with the regulator to find the regulator setting where the burner valve is most linear.

You might even need to find two settings of the regulator (not usually recommended). Maybe when you're using the burner valve between 6 and 10 you need the regulator pressure at 20", and when you're using it between 1 and 4 you need it at 10". But try to do the fine control with the burner valve. Regulators aren't really designed for this. (I took one apart because I wasn't sure the mechanic was right about how it worked - they're very interesting inside, an elegant design, but based on balancing spring tension and atmospheric pressure.)