I've never used a CFC and pour my wort from the kettle, so I'm not sure how important the whirlpool is. I find with my method, letting it settle is just as good as whirlpooling. However, the hop bag really does greatly reduce hop sludge in the kettle. I bet you'll be fine without the whirlpool.
What scares me about leaving the hop bag in during cooling, is that the hops will hold on to more wort at cooler temps. With a big hop schedule it could really hurt, especially with whole hops. Ideally I think it would be best to pull the grain bag just before flameout so it drains at the hottest possible temp. I'm definitely going to do the flameout addition outside the bag on the next batch, but it might be worth doing some experimenting to see the difference. Maybe a couple batches split before the boil.
I'm bottling a DIPA today that I used the hop bag method on and a mix of whole and pellet hops. By the time I got to flameout the bag was so full that it was hard to get the final addition below the top of the wort (Now I use a bag that the whole kettle could fit into). I let it sit for like 5 minutes before cooling, but I'm afraid that the last addition probably didn't do a whole lot for the beer. Unfortunately, I also dry-hopped the crap out of this sucker so I guess I won't be able to tell.