Its possible that what you are suffering from is ‘work hardening’ of the stainless. As you say a Carbide tap might solve the problem but when I have had the same experience I have always managed to complete the tapped hole by using several taps.
I presume you are using a taper tap, if so change to a second or plug tap, you will find that you can probably advance another turn before that tap too gets stuck, when that happens change taps again. By repeating the sequence you will eventually get through.
Another thought,,, do you really need a 10mm length of full M4 thread? Either counterbore one or both sides or use a larger tapping drill. It will still be a very strong fixing.
I have been told that in some cases cutting a thread from scratch with a plug tap actually requires less torque than the conventional taper/second/plug method. It certainly works for me in aluminium.
Ian P