I'm a bit wary of wading into this never ending debate, but I have a couple of observations.
First, we're forgetting that there is big variation amongst both fixed pitch and constant speed props. A friend at KLVK bought a constant speed prop off of a wrecked plane and its performance was so bad he switched to a Catto fixed pitch and it dramatically improved the performance. The constant speed prop was designed for an airframe that didn't have the performance envelope of RVs.
I have another friend who is helping Catto test designs of his new props for Long EZs. He uses an old Catto prop as his normal prop. We fly formation regularly and we flew together with one of the first iterations of the new prop and he immediately had to go back to Catto because he couldn't keep up with me in his Long EZ and was extremely disturbed
With the old Catto prop he is significantly faster than I am. So there is great variation even amongst fixed pitch props from the same manufacturer.
So switching from a bad version of one kind of prop to a good version of the other will be better, no matter which direction you're switching.
Now, going from props optimized for RVs, the fixed vs CS debate becomes much closer to religious arguments and I will bow out at this point