Found it. It's other tabs, not the tab you're typing in.
Compose drafts are saved to localStorage on every keystroke. Browsers deliver a storage event to every other open tab on the same site, and image_upload.js listened for all storage events. So each keystroke made every other ZenMemes tab with a composer re-check image eligibility, which fetches /api/get_profile. With a few threads open, that's a few requests per keystroke. It also briefly greys out Attach image in those tabs.
The listener only needs to react when the identity changes, which means the forum_pki_* keys or a full clear. That's the whole change:
https://github.com/gulkily/v3/pull/10
I've only syntax-checked it, not run it in a signed-in browser. To confirm after deploying, open two thread tabs, type in one, and watch the network panel in the other.
Checked the deployed fix in a real browser, since the earlier reply only syntax-checked it. Headless Chrome, two tabs on /compose/thread, a valid OpenPGP key in localStorage. Before the fix, typing 11 characters in tab A made tab B send 11 /api/get_profile requests. On the live build it sends none.
One smaller version of the same thing is still there. image_upload.js also re-runs the eligibility check on window focus and on visibilitychange, so the server gets a profile lookup every time you switch back to a ZenMemes tab with a composer open. That's per tab switch rather than per keystroke, but it's still a signal about when you're looking at the site. Throttling those re-checks to once a minute would fix it, and posting wouldn't be affected because the check before submitting calls refreshEligibility directly.