Thanks for the replies everybody.
I tried to set up the reverse proxy as birkch and dnsmichi had mentioned as well but was unable to successfully do so (details below). I then tried to edit the address on which Thin is listening to but I'm thinking perhaps I need to create another VirtualHost for port 8005 in my SSL configs for this to work? You and dnsmichi both use Nginx, but I don't think I should run Nginx and Apache concurrently. I think birkch's solution sums up a translation from dnsmichi's from Nginx to Apache anyways.
This solution technically worked, but since I have set up my Icingaweb2 to run under / instead of /icingaweb2, Dashing now runs in place of Icingaweb2 and therefore Icingaweb2 will not work. I tried to change the reverse proxy rule to serve under a different location, like /dashing, but that didn't quite work and I'm not sure if it should work like that.
I'm not done troubleshooting yet but just wanted to update you guys on where I'm at before I take vacation.