Before 1800, any profile with two given names is a red flag, and I would certainly count "Joan Alice" in that. Rather than detaching Joan Alice and leaving her to cause problems for future people trying to clean up WikiTree, I would suggest that you rename her Alice Unknown, consistent with ODNB.
RJ's research adds credibility to the thought that someone took the real Alice unknown and conflated her with Joan Briwere to produce Joan Alice. There's already a Joan Briwere, so renaming Joan Alice into Alice Unknown would finish undoing the conflation, and you'd end up with a profile for Alice which actually has a credible source!.