I've just tried to fix this myself using the Firefox add-on Greasemonkey. It allows you to make a script that runs when certain conditions are met, including visiting a certain web page.
If you install the add on then make a new script, and have the following in it.
// @name Outlook.com edge prompt
// @namespace outlook.com No Edge Prompt
// @description stop the outlook.com prompt to try it in Edge when not using that browser
// @include https://outlook.live.com/owa/
// @version 1
// @grant none
document.getElementById("ms-bgc-tp").style.visibility = "hidden";
I can confirm it does not seem to have broken anything else about outlook.com webmail.
Be warned that if Microsoft ever updates the web page and the CSS id for that ad changes, this script will stop working.
If anyone knows a better solution, I'm open to improvements/corrections.