Hey Matt,
Thanks for reaching out! I’m sorry to say that the specific thing you mention here is not possible at this time:
Is there a way to have a category of events use a particular pages template settings.
However, you can change the general events template and this may help a bit with the situation you describe.
To see if this helps at all, head to Events → Settings → Display in your wp-admin and try different settings for the “Events template” option.
Do any of the templates help bring the menu display to something like you describe? If there is progress with any template, let me know what the template is and if it comes from your theme, for example—then a bit of extra custom coding on that template may be possible to help selectively show the menu based on category.
I unfortunately cannot promise success here, but it’s possible! Let me know what you find by trying different settings for that option.
Cheers,
George