For everyone having this issue, I was able to find a workaround. Download Xcode 15.4 (there are a number of sites around the internet that host the old xip files). If you are already on Sequoia, Xcode 15 won't run out of the box, but you can open the app bundle, find the Info.plist file, and either overwrite it with the one from Xcode 16 or 16.1 or you can just hack it to make it think it's version 16. After that, Xcode 15 will run and you can make the changes to the Package Dependencies there. All the Bitbucket Cloud integration worked fine in Xcode 15. Then, just save and reopen the project in Xcode 16 to build and it should all work fine. Frustrating, but at there's a bit of a workaround for now until Apple fixes this.

Incredibly frustrating. I've recreated App Passwords in Bitbucket Cloud, granted every permission possible and I still just see that frustrating triangle of failure next to BitBucket Cloud when trying to add a remote package to my project. It says "Resource not found" when you hover over it. I am currently completely unable to change any package inclusions that involve BitBucket repos.