Do you mean you want to connect to the remote ssh server using an
https proxy? Or are you already connected to the remote server and
want to reconfigure your web browser to use a proxy? If the latter,
that's a job for your web browser; maybe try restarting it to see if
its "automatic proxy configuration" does something different when the
VPN is connected.
sshuttle can capture and redirect DNS requests if you use --dns,
although nowadays some browsers try to use dns-over-https or
something, and we can't redirect that.