Dyeing is tricky. I don't know what the fiber content is of this fabric, but a lot would depend on that. Natural fibers require one type of dye and synthetics require a different type. Don't even think about using Rit dye! It's not very lightfast for one thing, and you really need to use a hot dye bath for it to set. It will rub off on clothing if you don't set it.
Textile paint is a better option. It will stay soft when it's dry and will hold up to use. You can always touch it up if it wears off over time. Some people use latex wall paint and end up disappointed with the fabric is scratchy and rough---or worse, when it flakes off.
I don't know what the spray product is that you've heard about---whether it's dye or textile paint.
None of what I've said has anything to do about the advisability of altering the fabric on a chair of this value. I'm gonna say outta that one.
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