I am with Alan. I never use the strap. Not only do I find the straps to be awkward in securing a good playing position, I just don’t feel connected to the instrument. As you have found, using the strap can be tiring and painful to your fingers after a long session of playing.
When I use the strap, the instrument always feels like it is out of control, especially since I tend to play hard, often banging the keys for various effects. (I grew up playing rock Hammond organ, where if you hands were not bloody at the end of a show, you did not really the play the instrument.)
So I mostly hold the instrument with my left hand at the water valve end. This also lets me play a couple of high notes with my left hand for various effects.
I hold some melodicas by cradling them in the palm of my hand between my thumb and forefinger. You can twist your hand to have either the thumb to the right or left. You can achieve perfect balance with any instrument and really feel you have control of the instrument.