I believe the BrowserTools MCP tries to solve this:
It works, but is somewhat convoluted. You need 3 separate components running for it to work.
1 - Browser extension, 2 - the MCP tools and 3 - the BrowserTools Server.
Once that’s up and running, though, development with Cursor being UI-aware is pretty seamless (except for the massive, annoying banner on ALL your browser windows informing you that BrowserTools is running, which, after performing three separate steps, you’re already acutely aware of!)