Sounds like the new bike wants to fall in quicker than the SV. So guess what?
When you deliberately counter steer, the bike is going past the lean angle for that particular speed Thus it feels like it's falling. In effect, you are over countersteering because you now get more effect than you used to for the same movement.
No need to adjust anything, just don't do it
Counterseering should be a sunconscious act. It should be a part of a whole. Not a deliberate act. No need to deliberately over do it.
If you're looking for something new, try steering the bike using the balls of your feet. That's got a great zen like feel. Put more weight on your right foot to countersteer left and vise versa.
Think nuances, not actions. less is more. "Will" it to happen, don't do it.
Cheers, Carl