Hi Tim,
Yep, we support this. See https://github.com/moodle/moodle/blob/777d00ef37e8588a2ba74cea302e9553c2f0e85d/mod/lti/locallib.php#L3899.
If you configure the tool with a custom param, like so:
usertimezone=$Person.address.timezone
you'll see this come through in the 'https://purl.imsglobal.org/spec/lti/claim/custom' claim, under 'usertimezone'. This will be of the form "Australia/Perth". Of course you can name it whatever you like. I just used 'usertimezone' as an example.
One important thing I've just noticed, and which will need fixing: Currently, if your user profile is set to "User server default", it'll be sent over to the tool as "99" (the select option value we use for "server default"), instead of first being resolved to the proper string (like "Australia/Perth"). If you've set a specific timezone, it'll come over as that, however. I'll create a ticket to fix this, since 99 is meaningless. I've found MDL-76400 which deals with this exact issue.
Cheers,