Thickening and/or slowing evaporation of IPA-based ink
This is more of comment than a solution. 

It has often been noted here that chemistry is an experimental science so often you just need to try things. 

There are a number of considerations here. It seems like it takes on the order of 10 minutes to stamp a batch of "strips" (whatever that is...) and that:


You need a stamp with multiple "stamp faces" to stay wet that long. 
You also want the ink solvent to "dry" so that the ink solvent doesn't leave a noticeable "watermark" on the paper. 
You want to use a minimal amount of solvent so that the dye stamp doesn't bleed into the paper. 


(1) The dye must be fairly polar to dissolve in a solution of 50/50 mix of isopropyl Alcohol (2-propanol)  and distilled water. You can look at a table of solvents with their polarities listed at this webpage. You can try other solvents that you have, or mixtures that you can make, to get a rough idea of how polar the solvent (mixture?) must be. 


Reasonable to assume about 1 gram/ml for any of the solvents listed. Estimate polarity of a mixture by assuming ideal solution, ie that total solvent polarity is equal to molarity of a solvent times its mole fraction. (ie 50 wt % is not the same as 50 mole %.) 
You can check to see if the ink dissolved by seeing if the solution is "clear" - in other words shine a bright light through it. If you see scattering then the dye didn't dissolve. 
You can probably estimate from your work with glycerine how concentrated the dye must be to be invisible when dried, but visible under fluorescence. (I'm guess that the dry dye itself is green. So you don't want to "paint" the surface with particles of the dye. Rather you want the dry to be carried into the fibers of the paper as the solvent carrying the dye is adsorbed.)


(2) Look at other solvents that have been used for inks, for example on this webpage. Choose some solvents to test which have a polarity that works from step (1). Now look at solvents vs vapor pressure. The lower the vapor pressure the slower the solvent will be to evaporate. 


Note that a mixture won't evaporate homogeneously. So with an acetone/glycerin mixture the acetone evaporates preferentially to the glycerin. 
There is also an effect here in that the ink solvent is adsorbed into the paper not just sitting on the surface of the paper like a drop of such solvent on glass. So the correlation between vapor pressures and drying time would be "good" but certainly not perfect.