I understand that you think this, but I'm telling you (for at least the 4th time) it will be much more of a problem than you think. In fact, I'd go so far as to say it'll be your main problem. I know because I've been working on graphene transfer (supported and unsupported) for several years now.
You've also ignored my other questions: why not measure mobility with semi-supported graphene over a substrate with troughs? Have you tried simply immersing graphene on any substrate in Novec? Does that change mobility? When you etch the copper growth substrate, how are you going to get rid of the etchant? Won't that require handling the unsupported graphene? How are you going to know that some of the etchant (or residual copper ions) isn't sticking around and affecting your mobility (etchant is usually either APS or FeCl3)? Won't that require an elemental analysis (and thus handling the unsupported graphene)? What happens when you dilute water with Novec? If they're miscible, do you have a way to mitigate the Marangoni-like convection at the liquid surface? If they're not miscible (which I strongly suspect they're not) then you'll have two phases: water at the top and heavier fluorocarbon at the bottom. How will you remove the water then? How will you make a good electrical contact with the graphene? You keep using the phrase "pop onto the leads." This sounds like handling the unsupported graphene. If there's any popping (or any fast motion whatsoever), the graphene's going to crack or fold onto itself.
I don't want to discourage you from doing science along these lines, but 1) this is a far more difficult project than you're making it out to be, 2) you seem to be ignoring what the difficult part is, even when an expert with several years' experience is telling you what it is, and 3) you don't seem to have considered vastly simpler experiments which would inform you whether it was even remotely worthwhile to pursue the more difficult experiments. It's also unclear what problem you're trying to solve with this type of measurement. Are you simply trying to observe the effect of Novec on electron mobility? This can be done on a substrate with troughs far more easily. You say you want to use this setup to eliminate parasitic leakage (I'm not entirely sure what you mean by this): have you observed parasitic leakage in graphene supported on an insulating substrate (like SiOx or polyethylene)? Is it a big enough problem to warrant such a complicated solution? Have you searched the literature to see if someone has addressed the issues you'd like to solve?
At any rate, I would suggest that if you're going to play with this, you at least do two things: 1) clarify exactly what it is that you want to learn about the system, and 2) start with a much simpler system to determine if you need to go more complicated. I wish you the best of luck.