Probably not the advice you are looking for, but I know a lot of people use a cotton bottom sheet with a linen top sheet just for this reason.

If you want to decrease your price per week you could also shop sales. Discount stores like Nordstrom rack, HomeGoods, Marshalls, tj Maxx, etc sell discounted linen sheets. I just got a king size linen sheet set for $70 and if the bottom sheet lasts 2 years that is less than $1 a week. After that I can likely just replace the fitted sheet if I decide I want to stick with linen.

Otherwise I've read that the average time for a high quality (thicker) linen bottom sheet is around 5 years if you take really good care of it (hang dry) and rotate it. I've heard good things about magic linen. Good luck! This just seems to be the nature of linen based on what I've read.