The X-UA-Compatible meta element only changes the Document mode, not the Browser mode. The Browser mode is chosen before the page is requested, so there is no way to include any markup, JavaScript or such to change this. While the Document mode falls back to older standards and quirks modes of the rendering engine, the Browser mode just changes things like how the browser identifies, such as the User Agent string.
If you’d like to change the Browser mode for all users (rather than changing it manually in the tools or through the settings), the only way (AFAICT) is to get your site added to Microsoft’s Copat View List. This is maintained by Microsoft to apply overrides to sites which break. There is information on how to remove your site from the compat view list, but none I can find to request that you're added.
The preferred method however is to try to fix any issues on the site first, as when you don’t run using the latest document and browser mode you can not take advantage of improvements in the browser, such as increased performance.