The Dolomites are chock full of full service huts/rifugios. I think it would be difficult to do a multi-day hiking trip without having the opportunity to hike or scramble up a few peaks.
There's a book called "Walking in the Dolomites" which is helpful in putting together hiking routes, but doesn't have scrambling info. Despite their stigma here, there are via feratta routes which might be worth looking into. I believe they are remnants of the wars. I'd bet many scrambles have a few ladder rungs here and there, or might even be considered via ferrata routes, so I wouldn't let the term turn you off from checking something out unless you are an absolute purist.
I wasn't there long and had a sick travel partner, so just hiked around a bit, but I hope that helps a little. There's an expat in Munich who has posted some excellent Dolomite climbing reports here, hopefully he'll pipe in with some specific recommendations for you. If you happen through Bolzano/Bozen check out Messner's Mountain Museum, it's in an old castle and is worth a visit. Have fun!