Copying theme options from test to live databases

I am working on a moderately busy dog rescue website : https://www.oldies.org.uk/
This has updates coming in all the time, a fair bit of traffic and a team of volunteers working it. At the moment it’s on 2.92, I want to take it to 3.3, and update the theme while I’m at it.

I can’t take the site offline for long while I work on updating it, its themes and various bits of custom programming that plug into it so it all works with the latest wordpress.

I can’t risk it falling over while I work on it, and I can’t tell the volunteers to cease work till I have it sorted as this is a free time project and I’m not going to get it done in an evening. Oh, and the public can also submit some content, so even if I could call off the volunteers, I couldnt’ stop the public submissions…

I copied the database and put in a new shiny up to the minute install at a test location so I can get it all together. This works for testing my custom stuff.

But I’m not sure what to do about all the theme options. I’d like to use a modern theme as the basis for my custom child theme, with widgets and options & stuff. But because the widgets all store content in the database, I’m not sure how to get everything set up and tested while my volunteers are still beavering away adding posts & updating pages, in such a way that I can go live as quickly as possible, without a long pause while I fiddle with checkboxes.

Can I copy the theme / widget database fields over when I am ready to go live? Is that just the stuff in wp_options, or is there themey stuff squirrelled away elsewhere?

Or would it work better to freeze everything briefly while I copy the wp_posts table into the test database, do an update then copy it all back to live?

I’m sure this must be a common problem, can anyone advise how they have handled it?