What DNS servers does your DHCP scope give out? That should give you some clues as to where to look. If your PC is given an external DNS address, then you're out of luck, but if it's an internal address then whatever has that address may have the capability to hold custom records.
And if it doesn't, maybe it's time to set up your own DNS server so that you can do stuff like this. It's a fairly simple process (says the network engineer who's never done it before). If you're running an Active Directory domain then you should be able to add the DNS Server role to your domain controller,
or look at BIND if you're a Linuxy person. And if you've got a domain controller, you could also add the DHCP server role to it, and control things yourself that way, without having to fight with a shit router.
I would say leave the managed switch to do switchy things, though. Let it worry about moving packets around and very little else. It's not a normal place for a DNS server or a DHCP server.
Configuring a router for ADSL shouldn't really be that much of a big thing (the ISP should be able to help, especially if you're paying business rates), but I'd also say that you should maybe have a look at the local IT firms and ask them how much it would cost. They do that kind of thing all the time and chances are it'll come in cheaper than paying you to spend much longer working it out, and if not cheaper then at least you'll come out of it with a simpler setup. If you find the right firm, they should be able to set it up for you and provide proper instructions on how to make the simpler changes without having to go back to them.