Try a different USB port. I had an issue where DCS was sensitive to the port structure (I think that it had something to do with power and whatever libraries DCS uses to access the joystick seems to be somewhat sensitive to this). When I switched the joystick to a different port, the problem went away. I ended up buying a powered USB hub to attach all of my devices to.
I have an X-56 Rhino, but I don’t use the Logi/Saitek software beyond calibrating it and such. I do all my mapping inside DCS.
I don’t know if my way is the “harder” way (since it certainly doesn’t feel easy mapping things for each and every plane and helo), but I’ve never experienced what you’re describing. If that’s because there is some flaw in the profile software, then I’m glad I’m not using it.
Of course the flaw could be in how DCS reacts to the software which maybe works fine everywhere else, I don’t know.
I know for certain when the KB gets into “spam mode” it is repeating whichever key so fast it can ignore user keystrokes. I am not sure if this would affect other inputs, as I have only had it come up when unlocking my PC.