Update: I found a partial solution using Nitter and the Link Gopher Firefox extension.
Using a public Nitter instance, one can visit an account page, jump to the bottom, click “Load more”, and use Link Gopher’s “Extract Links by Filter” tool to extract all links of the form https://nitter.freedit.eu/SpaceX/status/. This method allows one to collect 20 links at a time. It’s still a bit more work than I would ideally like, but better than trying to find and copy each tweet URL manually. Also not sure how much longer this method will work given that public Nitter instances will likely stop working soon.