What I would do, in your shoes, is get swatches of the types you're considering and test dyeing them. In theory, RIT should work -- linen is a natural plant fibre much like cotton, and can be dyed in a similar fashion, though the way the dye takes may not be the same. (I can't speak to personal experience with RIT dye on linen -- I've used fibre-reactive dyes in the past myself.) The important thing with dyeing is to make sure there's no finish to the fabric, and if the fabric's been softened that may alter how it dyes -- but at least your store looks like they have swatches, and that's a definite plus.
Good luck! Hope that helps some ~
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