Powershell context menu shortcuts such as CTRL+M don’t work

I am not using TMux, but I suspect you need to verify the Ctrl-Shift setting of Mintty

If there are multiple context menu items with the same hotkey, pressing that hotkey will toggle between them as you discovered. If a hotkey only corresponds to one item, pressing it will immediately trigger the item.

Evidently, Windows is using the first letter of the item's text as the hotkey. You can override that by placing an ampersand (&) before the letter that you want to become the hotkey. However, this must be done in the default value of the command's Registry key, not the key name. In this image, I've tried to change the hotkey for Contemplate to O and the hotkey for Edit to I:

Both appear to work in that the above configuration produces this context menu:

Pressing O immediately triggers Contemplate, but pressing I does nothing even though no other item in my menu uses I. Changing Edit to have an ampersandless key name but a default value of Ed&it makes it work as expected.

There is not a straightforward way to change the built-in context menu items like Copy; they are not in the Registry and are instead generated by Explorer. You might be able to write a shell extension that uses SetMenuItemInfo with appropriate menu information structures to change it at runtime. For an older answer, I wrote a shell extension to change built-in items' icons that might be a good starting point.