Thank you for the suggestions @EA_Leeuw and @puzzlezaddict
Unfortunately I think I've been through everything now and nothing's worked.
Here are the steps I took, with no success:
1. Used Revo Uninstaller to uninstall TS3 and EA App, using the Advanced option to remove all related files.
2. Restarted my computer.
3. Quit Malwarebytes, and turned off Windows Defender firewall and real-time virus protection.
4. Re-downloaded and installed the EA app.
5. Ran EA App Updater. Learned I do indeed have the latest version.
5. Ran EA App as administrator.
6. In File Explorer, created file C:\\Games
7. In EA App, chose to install TS3 base game and various EPs to my folder C:\\Games, resulting in the actual install location C:\\Games\The Sims 3
8. Observed the base game installed okay. At one point I got an error popup, but no red text beneath the progress bar, and the base game installed and launches fine. However, when it came to the EPs, it was the usual "Something went wrong. Try again later," and inability to resume download.
9. Ran EA App Error Reporter and submitted a report.
10. Tried once more to install an EP on top of the now re-installed base game. I had to manually choose an install location, to which the EA App added that weird second dash. And when I confirmed I got the familiar red text, "Something went wrong. Try again later" beneath the progress bar, and the download was unsuccessful.
(11. Obviously, re-enabled my anti-malware.)
I think all of that included everything suggested so far.
The Error Report reference I received at step 9 was 559c49f3-5c62-45f0-9a3e-0899c215f5bc