Here is how you actually determine if sucrose from beet sugar is identical chemically to cane sugar.
1.  The first experiment would be to determine the optical rotation of each.  When you do this you will find both have an optical rotation of 66.37 degrees in standard units.
2.  The next experiment would likely be to determine the molecular composition of each sample.  This is easy enough to do by simply doing an exact mass by mass spectrometry.  You will find both beet and cane sugar have the empirical formula C12H22O11.
3.  Probably the next logical experiment to do would be to compare the infra red spectra of samples from both sources.  The easy way to do this would be to prepare a pellet of the sample mixed with potassium bromide.  As nearly all organic chemicals exist in more than one stable crystal isomorph you also need to make sure both samples have the same crystal structure.  The easy way to do this would be to take both and make a mildly super saturated water solution at room temperature and add a tiny ground up seed crystal prepared from one of the sources.  Make that seed small enough so it contributes no more than one part in ten thousand to the final crystalline product.  Filter and dry the resulting crystals, make the KBr pellet and run the IR spectra and you will find sucrose from both cane and beets give identical infra red spectra.
4.  This is really becoming over kill, but another dirt easy experiment to do would be to run high field proton and carbon 13 NMR spectra on samples from each source with the sample dissolved in D2O.  When you do this you will find both the proton and carbon 13 spectra are identical.
5.  If you want to go to ridiculous over kill you could treat samples from each source with the enzyme invertase and show by liquid chromatography that both produced equal molar amounts of fructose and glucose, both of which you can buy in pure form from chemical supply companies to use as standards.  This is more work than all of the above experiments combined.
6.  Equally ridiculous over kill would be to take a sample of each from your recrystallization scheme in step 3 and grind up the sample and run an xray powder diffraction pattern which would show both samples give identical diffraction patterns.

All of the above experiments have been done and are reported in the chemical literature along with many other experiments that show both sucroses are identical chemically.  Your bees will do just fine on sucrose from either source.  In fact the only experiment you could do to figure out if the source was beets or sugar cane would be to determine the exact ratio of the carbon 12 to carbon 13 isotopes in your sample.  Carbon 13's natural abundance is about 1.1%.  But beets and sugar cane use slightly different biochemical pathways to make sucrose and this difference results in a very small difference in the carbon 13 abundance between the two samples.  Nearly all nectar sources use the identical pathway used in sugar beets.  So, if you are set on mimicking a honey bees "natural" diet you should use sucrose from sugar beets.   But, I can assure you that your bees will not care a bit which you use as they are exactly the same chemical.

Doing some cooking experiment to try to determine chemical identity is an idea that was out grown about 1820.  We study chemistry in school today, not alchemy.

Dick

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