Not knowing what the pram looks like it's difficult to give advice, but personally I wouldn't use a fabric dye for a couple of reasons. One is, will the dye be harmful to the baby? And the other is, will the dye rub off on the baby's skin or clothing?
If at all possible, I would suggest recovering with new fabric or there must be fabric cover-ups like they have for car safety seats, chairs, sofas, etc but made for prams.
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