You'll need to restore the mbox file underneath 'Local Folders'. Then copy the messages to the desired folder on the server. Don't use drag'n drop. Instead select the messages to be copied, and right-click a selected message - Copy To. Once the upload completed successfully you can delete the local copy. Don't try to upload all messages at once, do that in batches.

Unfortunately, that’s not a great solution for me, as this mbox has 200,000 messages in it. Is there another approach that would work?

I dragged the mbox file into the gmail account in Thunderbird, and the messages immediately started uploading.

I'd expect the server to right away delete the local messages you pasted into the Gmail account. This is how IMAP works.

But after a few minutes, I got an IMAP timeout error message from Thunderbird.

Did you manage to upload any messages at all this way? If so, where those messages still visible in Thunderbird?

Is there another approach that would work?

None that I'm aware of.