Add https://gist.githubusercontent.com/programmerq/6a6c127a94c9018f1a0503fa0140770a/raw/custom-icon.js as a user script using your preferred userscript extension. I used it with TamperMonkey on Chrome with success.
If your Teleport instance is not at teleport.sh, add the Teleport cluster URL as a user match.
To set a custom icon, base64 encode a 72x72 png image, and set it as the value for a label called teleport.dev/icon
.
Resize and/or convert to png if needed. Many logos can also use a reduced color count: