Data point: You can get both the AF version and the free version at the same time and on the same day.
My sister has to stay at a Days Inn soon and wanted to not pay cash. Needed both cards to get enough points for the stay. Didn't do a classic 2BM, but applied for 1, got instantly approved, and then applied for the other (in a different browser). The site said she has an app in the system right now (so we're not sure whether it even went through) but she called up, they said we have to pull your credit because it didn't pull before (which implies to me that the app didn't go through) and then she got approved after recon! Hopefully both cards will result in only 1 pull on her CR as they took place within a few minutes of each other, but as people have been reporting in the Chase thread, it might be 2 pulls these days.