There are a couple ways to handle this.

1. This happens with PowerDeleteSuite's main version, but one of the forks has a 5 sec pause between deletes that does work. It's very slow (~1000 comments edited in 2 hours), but works reliably.
   https://github.com/deestan/PowerDeleteSuite

2. This userscript will manually overwrite comments manually. You go to your comments page, scroll until there are no more comments to load, and run it. It'll go 1 by 1 - edit, new text entered, save, 3 sec pause. By default the replacement text is a link to the script, but you can just edit that to be whatever you want (and you should edit it to make it harder for Reddit to batch restore).
   https://greasyfork.org/en/scripts/468337-so-long-reddit-thanks-for-all-the-fish

Note 1: Method #2 doesn't use the API, so should still work on July 1. PDS does use the API, so might not.
Note 2: Both methods allow you to continue to use the browser in another tab or window (at least in Firefox). Some browsers and extensions will suspend or unload tabs if they haven't been viewed in a while, so best policy is to do this in a separate window.
Note 3: The default PowerDelete Suite should be fine for exclusively deleting comments if you run it several times. Just keep running it until nothing shows up under comments. To be sure, once your comments are all gone, wait an hour, and check again. If anything remains, use method #2.

(Overwriting account #7 right now! Using PDS with 5 sec pause has worked flawlessly.)

Pass this along to anyone having trouble!