I know this thread is a few years old, but it's still relevant for anyone searching this now. We went through something similar and ultimately just walked away from a designer who wanted the same kind of "property until construction contract" clause. It felt like too much leverage on their side for a $125/hour fee, especially since you have no visibility into what the construction contract terms would even look like.

What worked better for us was finding someone who charged a flat design fee upfront and let us own the plans outright, no strings attached to using them as GC. We ended up going with independent designers rather than a design-build firm for exactly this reason, it kept our options open for bidding out labor separately. If you're in a market with a lot of design-build shops (like some of the searches I've seen for kitchen designers in connecticut), it might be worth specifically looking for ones who advertise separate design and build contracts, since that structure tends to avoid this whole ownership issue from the start.

Either way, I'd push back on the exclusivity clause before paying anything. Even a modest change, like plans becoming yours after a set waiting period if you don't sign with them, is a reasonable ask.