I love the bidet idea. Other than that you've got an array of tissue quality options that can impact price significantly.
You could try to go for the truckstop/restaurant quality giant rolls, might save you a bit, but standard sized rolls frequently go on sale. The Costco toilet paper standard price (might vary by area) is $0.00125/sheet. If you already have a membership that's a great option. Otherwise stock up when there are good sales. One of the foremost experts on toilet paper deal tracking only purchases when the price is under $0.00253 per sheet, but few sales strike lower than the Costco price, just be sure to factor in membership cost if you don't already have one.
Using $3/week sounds like a lot. Maybe you have a large household and this reasonable, but I'm guessing there is a technique issue. Apparently a large number of people just tear/wad/wipe and repeat. Even https://howtowipeyourbutt.com/ shows the inefficient wad method. The more efficient approach is wipe, fold, wipe again, until the the folded size becomes unusable. What I've gleaned from personal experience and quick scan of some search results is that people who fold typically use 3-4 sheets to clean themselves up.
There could also be a dietary component, if every cleaning effort is an ordeal that is not normal.
Edit: Just checked Sam's Club, they have 1,000 foot rolls of rest-stop/restaurant style 2-ply 6 for $20.34. These are non-perforated, but if you go with 3-sheets per foot that's a per sheet price of $0.00113. Not a pleasant way to save a few cents a week in my opinion.