Yeah, don't be that guy. You don't want to show up at a school asking for a fight. Even if you make it perfectly clear it's "just sparring", many people will see it as a challenge, and you're going to get a lot of angry looks and possibly get hurt.
As I understand it, you're interested in an exchange of knowledge. You're not looking for a fight. You're looking for like-minded people who want to share what they know and see what others know.
Most people are not as open minded as you. And most instructors won't care to teach you anything, because they're busy with their paying students. But there should be a good amount of interest out there for exactly what you're interested in.
When I was a university student, some of the black belts from various martial arts clubs would get together once a month or so outside of their regular classes to put on a little mini workshop for each other. We called it, "Black Belt Club". Haha.
The topic could be anything. For example, I once instructed a class on Judo breakfalls. We would rotate through different people (all black belts in different styles). Each month, someone else would be the instructor. Everyone got to try a sample of what the other martial arts were doing.
So that's what I recommend. Find a bunch of advanced level students or instructors in different martial arts. Meet at a neutral place on some set schedule. Emphasize that the purpose of this is to exchange ideas and learn. Make it free, unless you have to pay to use the facilities somewhere.
Your main problem will be finding a location that has mats, if you need them. We would meet at our university's wrestling room, which had mats already in there. You can try renting out floor space at a gymnastics school during off-peak hours.
I'd look into getting some legal waivers ahead of time, by the way.
Hope that helps.