v3:
url format = chrome-extension://aglmepmfibmokigfgakpbcimgkkkklia/_favicon/?pageUrl=https%3A%2F%2Fvideo.twimg.com&size=64
Screenshot from 2022-12-02 20-01-04

v2:
url format = chrome://favicon/size/32@1x/https://google.com/
Screenshot from 2022-12-02 19-07-30

but v2 is forbidden to be published.
I don't know how long they're cached nor do I know how to tell the browser to cache it longer. Would you be able to inspect the image with JS in some way to determine if it was a cache miss, and then implement your own fallback? This is a possible workaround.
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"January 17, 2022: New Manifest V2 extensions will no longer be accepted by the Chrome Web Store."