I suffer from an unstable idle setting on my BME 44. Having set up the engine to a reliable idle with the transmitter throttle trim, I have a successful flight with good high end and transition. When I start the engine for a second flight, the idle is too high and I have to lower the trim. The next time, it is too low, etc. I have to continue to make this adjustment after each flight.

I have already removed the carb, and reinstalled using silicone gasket paste.

Other than this constant idle adjustment, the engine runs great, with lows of power.

Any suggestions would be appreciated.

There may be a couple of reasons for this.
First:
by preference, use not the transmitter for lowest idle trim, but the idle set screw on the carb, and switch off the engine using the ignition kill switch. This allows for a very reliable idle setup. For a faster idle, you can use the transmitter trim. This system is used in motorcycles and cars as well, and is very reliable.

Your problem can arise from either a slightly binding throttle pushrod, or a critical setup of the bellcrank angles around idle. Set up the carb belcrank to pushrod angle to a maximum of 45 degrees accute angle, with the servo bellcrank almost in line with the pushrod. This gives a very stable idle position. The remaining servo travel should just be able to kill the engine.
Setting the carb belcrank more accute, like 30 degrees, increases the effect of pushrod play or binding, causing the problems you describe. Even a slight temperature change or play has a large effect on the effective length of nyrod pushrods. Since the carb lever is the rod's free end, idle is bound to vary.

With a good servo and proper linkage you should not have to use the engine idle stop screw, if using the screw does help it is an indication that something in the servo/linkage is not always returning to the same position, it is always nice to be able to stop the engine by moving the idle trim lever to full down position in my opinion.

I agree, that is why I said: "by preference"
Stuntman has not indicated his exact setup, so maybe not using the idle screw causes his problem, and setting the idle by the screw may help him locate it. One important variable less to cope with.