Client TLS is rather unusable on the Internet by a typical random end user visiting a random public site, so that should at least keep the specific scenario you describe at bay.
Of course the website itself would need to support that, but it's all possible in time.
https://privacysandbox.google.com/blog/update-on-plans-for-p...
I suppose client cert would protect against from a MitM attack, if the client failed to notice it, or if the MitMer has the website keys to make a perfect attack.