Ah. There's no way to convert cookies. I mean, I guess you could proxy all FusionAuth access through an NGINX instance or something like that, in which case you'd control the cookies, but there's no setting in FusionAuth to modify the cookies. Feel free to open a feature request referencing this forum post if you feel like it'd be a good feature.
For the present, you might be able to make a GET request against the logout endpoint in the window.close event listener.
A bit of googling turned this up: https://stackoverflow.com/questions/6162188/javascript-browsers-window-close-send-an-ajax-request-or-run-a-script-on-win
Let me know if you experiment and find a way to do this; I'd be interested to learn what works.