Some additional info. I changed my theme to both Astra and GeneratePress in staging to try to confirm if this is a bug with the Kadence theme. The same behavior (the addition of the BR tags) appears in those themes as well–but, again, not with Twenty Twenty-Three.
Also FYI: I had this working as desired (no BR tags) in Kadence within the last week or two.
I found the issue. It’s some sort of conflict with the plugin Popup Maker:
Popup Maker – Popup for opt-ins, lead gen, & more
The plugin includes a targeting feature that I can use to exclude BP pages. When I apply that filter, or turn the plugin off entirely, the problem goes away.
Sorry for the clutter!