You could use the media-fragment syntax in the url to let the browser know which portion of the media it should play.
For instance, to tell it to play only the 5 first seconds of your media you would set your url to
https://www.example.com/example.mp4#t=0,5
document.querySelector("button").onclick = ({target}) => {
target.remove();
document.querySelector("video").play();
};
video { max-height: 100vh; }
<button>play</button>
<video muted src="https://upload.wikimedia.org/wikipedia/commons/2/22/Volcano_Lava_Sample.webm#t=0,5"></video>
The browser may preload more data than just this portion, but it should give it enough of an hint to not preload too much. Also, beware that if you let the user play with the controls of the video, they can easily over-pass that range.