This is brilliant.
If you're willing to have Claude write code to test a thing you could do a teeny bit more work and make that Playwright script a permanent part of your codebase. Then, the script can run in your CI on every build, and you can keep enhancing it as your product changes so it keeps proving that area of your product works as desired.
Have it run inside a harness that spins up its own server & stack (DB etc.) and boom - you now have an end-to-end test suite!