I did some reading up on the Dream Machine - I am more familiar with Ubiquiti’s other products (USG, EdgeRouter), and unfortunately I’m not sure the Dream Machine supports what you’ll need to do here. It is a frequent feature request, but the other devices support dnsmasq or the like which allow local hostnames to resolve correctly. I don’t have access to a Dream Machine to look around, so if someone else does, they may be able to help. But what you’re looking for is more advanced DNS configuration.
Now if you really are looking for a project, there are plenty of tutorials on setting up a Raspberry Pi as a DNS server…