Unfortunately, there is no built-in way to change how the full screen warning works only for some websites. There might be an add-on which can do that.

You may already know about shortening the display time as a partial workaround. For example:

(1) In a new tab, type or paste about:config in the address bar and press Enter/Return. Click the button promising to be careful.

(2) In the search box above the list, type or paste full- and pause while the list is filtered

(3) Double-click the full-screen-api.warning.delay preference and set it to 0 (that's a zero) and click OK.

(4) Double-click the full-screen-api.warning.timeout preference and set it to a short time such as 250 (that's a quarter of a second -- 250/1000 milliseconds) and click OK. I'm not sure if you can find a time that works well for all purposes. Times around 50 barely show at all, while times around 500 are more noticeable for regular use but could be annoying on the game site.