I've noticed your question a bit later and wanted to help you out a bit. Unfortunately, as you could already notice, this feature is not in YouTube's iFrame or Embed API. The best possible way to get maximized quality from YouTube is to create a backend server that is able to serve streams and fetch video's data (youtubei.js module is like made for this, I use it a lot and it gives you basically access to YouTube's API that YouTube itself uses) and use the server to get highest stream quality. but remember, video and audio are separated streams, so you should consider merging those streams together or using different method to stream both of them at once. But if you don't really need the extra high quality, remuxed stream (usually has around 480p, has both audio and video in one stream) could be used too.

https://github.com/LuanRT/YouTube.js take a look yourself, browser version is supported too if i'm not wrong