Sure, a managed switch might very well have its own DHCP server, but why all this trouble for a named network? It's just a name, it serves no other useful purpose.
You could also install a DHCP server on one of the computers, as a service. I'm sure there are freebies out there. Or just grab an old router, something cheap, even something from the local thrift shop, and use it. I wouldn't spend big bucks on a managed switch just to get a DHCP server.