Which easy-to-acquire clear liquid would be best for testing whether or not a twenty-sided die is balanced?
You don't want a denser liquid. You want a liquid with just the right density so the dice would barely float, almost fully submerged. To this end, you mix the solutions of different concentration and so fine-tune the density, as is beautifully described in the accepted answer to your rpg.SE question.

A denser liquid is worse for your purpose. Ultimately, with a very dense liquid (like mercury) the dice would just sit on one face, much like they do on a tabletop.

In the relatively rare case your dice might be made of different kind of plastic and wouldn't float at all. That's when you genuinely do need a denser solution. Chemistry knows a lot of salts; which of these are available to you is another question. Saturated solution of $\ce{MgSO4}$ has the density of 1.3 or so; $\ce{CaCl2}$ will get you to ~1.4, and other salts farther yet, but they are also more difficult to come by.

As for the reference books with some density values, you have to read those really, really careful to see what they actually refer to, and then read again, just in case. 1.028 looks like the density of seawater; it is salty all right, but that's pretty far from saturation. The solution you use is a lot more dense. 1.44 for hydrogen peroxide looks like the density of the pure compound, which you surely can't acquire (which in turn is probably good for your survival); what you had is probably the 30% solution which is a lot less dense.

As for the density at which your d20 will float... well, how are we supposed to guess that? For all we know, your die may be carved out of wood; also, chances are it is cast in bronze. The resulting figures may vary pretty widely. Upd. So it is plastic, after all. Well, that still leaves a pretty wide room for wiggling. You can't tell the density until you measure it. Chemistry is an experimental science, they say.

So it goes.