Good for you if you want to do install 4 tires, all the better.
But I can tell you for having driven quite a few cars (FWD and RWD) with Winter tires on drivetrain only that it NEVER changed the handling one bit on normal, wet and dry pavement.
On ice and snow, as I have already written earlier, a RWD might behave like a FWD and oversteer (the enhanced rear traction prevents the rear end from fishtailing, "pushing" the front end which has less traction). Well, you test, you adapt your driving style.
I'm not saying it might not alter handling on
certain cars, I'm saying on a dozen different models I drove it never brought any noticeable change (MUCH LESS on a FWD).
(The same way mounting 2 non R/F in front on my wife former Cooper Clubman did not change anything at all in the handling, although everybody and his dog including the BMW dealer said it "would". Well it didn't.)
So fair enough for whatever these associations do recommend, but in my opinion their recommandations always seem to steer towards the most costly option which will bring extra biz to the aftermarket car parts industry.