I have a part shade garden too & also have a moss in borders problem. Like you it's between plants & I spend quite a bit of time removing it & dumping it into the council recycling bin, not my compost bin.
If I was you I'd try to 'skim' it off & dispose of in a recycling bin, if possible. I dont do hanging baskets, but somebody locally may be grateful for it?
Unless you can dig a deepish trench on some new ground to well & truely bury it, I would bag it up to dispose of how you will.
BTW like you I allow my lawn to be a moss patch. Am sure though that this doesnt help, as the spores will get onto the cultivated soil & there's our problem.
I have addressed some of the shade problem by having a large tree felled, but the last 'no sun lots of rain summer' summer didnt help. Good luck, no easy way. J.