The absorption speed of a paper depends on two variables: 1) the presence and type of coating on the outside surface of the sheet and 2) the quantity of sizing or binder (a sort of glue/filler) mixed in with the fibers inside the sheet.
Coatings convey gloss to the sheet and furnish a designed surface for the application of commercial printing inks; we will omit any further discussion of them here.
Almost all uncoated commercially available ("plain") paper has some binder in the sheet because this conveys wet strength. Papers with little or no sizing (chromatography paper, kleenex, paper towels, toilet tissue, coffee filter paper, early types of inkjet paper) are specifically made to be highly and quickly absorptive. As a result, they have little or no wet strength, as you have discovered.
Lengthening the fibers in the sheet helps improve wet strength in papers without binder which means specification of the pulpwood source and the amount of recycled stock used in the sheet (recycling reduces fiber lengths and produces a weaker sheet).
Chromatography paper is by far your best bet because it is specifically designed to do exactly what you desire and it is also far more consistent from batch to batch than any other commercial grade of paper. Review your cost model to see whether or not the paper cost will make a significant difference.
One of the most quickly absorptive, mass manufactured papers is the early inkjet paper I mentioned; this was specifically designed to be strong enough to be tractor-fed through an HP inkjet printer mechanism while having the fastest absorption time of any paper for printing use.
I do not know if it is available any more since the printer that required it (HP's original "ThinkJet" model from 1984) has been off the market for years. The paper was manufactured for us by Mosinee Mills.