As you can read from the links in this post, paper filters eliminate almost all of the LDL-raising diterpenes from coffee. The French Press method, considered to be the "ideal" way of making coffee, is thus not good as far as cholesterol is concerned, and metal (gold) filters also don't protect you from these diterpenes.
While you can rinse and reuse the AeroPress filters, the amount of diterpenes increases, per this post. (The test rinsed the filter after 10 uses, and it showed that a rinsed filter resulted in nearly double the diterpenes of a new filter; I don't know if rinsing after one pressing would show the same increase in the subsequent pressing.) Since the filters cost only 1 cent each, it probably makes sense to use a new filter each time, which is what I'll now do.
(This suggests to me that espresso, being unfiltered, is also not good for LDL levels.)