The tab key in BBEdit inserts a literal tab character, or advances to a tab stop. It doesn't indent the line, which is what I would argue it should do when writing code.
I think about this has having two separate but related concerns; how do we build tools that focus attention on the parts of the problem where human judgement is highest leverage; and how do we build harnesses that enforce invariants such that humans trust the attention focussing part. It's a challenge!
I keep my agent workflows distinct; we have (for historic reasons) a lot of non-git controlled context that differs between branches, and moving all that around on checkouts is untenable. I use this tool: