Welcome to the forum!
I probably can't answer that in any satisfying way for you. But I see that there are 3 other choices of units, besides degrees. I don't know math well enough to understand them. (Hah! Or more precisely, I don't remember math well enough.) But I wonder if any of them can be converted. Well I mean, I wonder if the minutes can be converted to any of them?
I don't know of any way to add minutes to the menu, except it's probably by hard coding the program. If you happen to know the right code, I'm sure developers can help you with whatever you need.
I searched new feature requests, but the only reference to "minutes" was about time and not angles. So you certainly could make a feature request about that. But who knows how long it would take for that to happen - if it would happen at all.
So as for now, I don't really know any way to accomplish angles measured in minutes.
Maybe someone else will post with a more hopeful reply for you. That's all I know or can quickly learn about it.
Oh.... Well I thought of something, but it's still not really what you want. You can increase the number of digits after the decimal point, which gets you incrementally closer. But for one third - I do remember enough math that you just never get there (using decimals). It's .3333 infinitely.
Wish I could be more helpful.