So say you don't have the file for the game's SR logo. Is there a way to get the file from the site? I tried right click and open image in new tab, but it's the smaller image file.

https://www.speedrun.com/promotion

No, the logo made for the game's page not the default one.

You can right click and open the image in new tab before downloading it from there, I believe.

Logos uploaded are resized to a maximum of 720×128. We don't store any original files higher resolution than that.

Before 2017-11-05 however, they were resized to a maximum of 180×32, so any logos uploaded before that are still low quality.