Bricklink has this information for most sets. If you search for the set number, you get several entries. The top one is the entire set, and has the weight for the full sealed box with parts and instructions. Then there are two more entries, one for the Instructions and one for the Box. See here for an example using set 41314 Stephanie’s House.
If you go into each of these entries, you will find a weight for each. In the case of 41314, the total set weighs 1340g, the instructions weigh 318g and the empty box weighs 210g. So, doing the math, the pieces alone (assuming any stickers have negligible weight) weigh 1340-210-318=812g.
I don’t think there is a clear percentage for the weight of the instructions compared to the weight of the set, but feel free to use the BrickLink API to extract the above 3 pieces of information for ALL sets and calculate the average. I think you will find that the percentage varies due to a number of factors, such as: the number of instruction books used, the quality of the binding (some books are glued, some have staples), the weight of the paper used (some instructions use thinner paper than others), the set theme (IDEAS sets tend to have long explanatory stories before they dive into instructions), etc.