How to know if acid is polyprotic?
As discussed in your question and ensuing comments, there are ways to predict via theory whether a particular compound would be a polyprotic acid, but your last sentence asks how to prove that an acid is polyprotic. This pretty much limits us, as you proposed, to laboratory experiment, namely acid-base titration.  

Although your suggestion "Acid base reaction with something like sodium hydroxide NaOH and then distillation to separate the product/products from the water that is produced" is on track, the experiment needn't be nearly so complex.  

The experimental procedure is as simple as preparing a solution of the unkown acid, then titrating with a strong base solution while monitoring the pH.

The following is a titration curve for the diprotic oxalic acid, using sodium hydroxide as the strong base titrant. (Image taken from this boundless.com post):  

  

Note that there are two equivalence points, at 25 and 50 ml of added titrant. These equivalence points are the features to look for when titrating the unknown acid. A single equivalence point indicates a monoprotic acid, two indicate a diprotic acid, etc.  

There are a couple drawbacks to this experimental procedure. First, the equivalence points are not always as well defined as that shown in the figure. For instance, two acidic moieties may have similar Ka values and thus may almost appear as one.  Making a derivative plot of the titration curve can help to distinguish the equivalence points. In other words, you plot the change in pH vs volume of titrant added rather than just the pH of the titrant added.  

The second drawback of this procedure is that it doesn't tell you which protons were titrated by the base. This would require a far more complex experiment, for instance by synthesizing the acid using deuterium in place of hydrogen at specific sites, then analyzing the compound after each equivalence point to see whether the deuterium was pulled off.  

I hope I understood your question correctly and that my explanation of the titration procedure to determine polyprotic acids was clear. Don't hesitate to ask for clarification in the comments.