Not really possible for several reasons.Great skin!
Small usability suggestion:
Would it be possible to simply left or right click to paste and enter? Maybe double click to prevent accidentally changing the video.
Rainmeter skins cannot paste from the clipboard; perhaps this can be done with Lua, or Send key in a script, but that is a bit overkill for a simple key press IMO.
This skin uses the 'watch' link and converts it to the 'embed' format. The 'watch' link is different from the actual video's 'page' link.
That is why I mentioned to use the 'Copy link' from the right click menu of the video thumbnail.
If you left click the thumbnail image, the entire video page opens in a browser window. That is also why the 'Copy video URL' or 'Copy embed code' from the playing video's right click menu are not viable links.
I don't see where you can inadvertently click to change the video. The Rainmeter skin actions are completely separate from the Webview video controls. The 2 items have no knowledge about the other and they do not interact.
But feel free to modify it to your preferences.

Statistics: Posted by eclectic-tech — 44 minutes ago