To control everything we seem to need 2 yoyos with identical shape and specs (incl. total weight) but different weight distribution…I suppose it is possible with one full aluminum and one plastic+denser metal ring, but lol.
Easy. One Drop Benchmark series. 
Oh, sorry, you said identical shape, too. My bad!
Checkmate and 7075 Checkmate have identical shape but mass difference of about 4 grams. Checkmate and Agonist have almost identical shape (just the rim is a bit different) and mass difference of about 5 grams.
If high rotational inertia makes an object easier to accelerate, that would make it less floaty, right? So yo-yos with high rotational inertia (yo-yos with high diameter:width ratio, D-bearing yo-yos) should not be floaty and the floatier yo-yos would be yo-yos with low rotational inertia, which is caused by more weight distribution towards the middle and a low diameter:width ratio. This seems to mesh with some people’s intuitive notion of which yo-yos are “floaty” anyway.
Here is an example of tech:http://youtu.be/3dphY4hFvp0
Note all of the string layers between his hands.