DHCP Server Issuing NAK when servicing multiple subnets

Woah, there. What you're saying contradicts itself. You say "single subnet" in one point, but then "VLAN each site" in the second point. Then you say "the networks will NOT be routed". Are you sure you know what you're saying here?

Typically 802.1q VLANs are deployed in a one-to-one relationship with IP subnets. Each 802.1q VLAN acts as an independent Ethernet broadcast domain and, as such, broadcasts from one VLAN (like, say, a machine ARP'ing for another machine in the local subnet) won't be forwarded between the VLANs. Splitting a single IP subnet across multiple VLANs requires a "smart" bridge that can do proxy ARP.

How are you planning to get ARP to work between these various VLANs?

If you really want to eliminate "cross-site 'chatter'" then what you really want is a subnet for each physical location, a router at each location connected to the "MAN" to route traffic to the other locations, and "ip-helper" functionality in each router to forward DHCP requests from the various locations to the central DHCP server.

What it sounds like you don't want is a single big subnet with a bunch of bridges running proxy ARP, in my opinion. Your DHCP inquiry really, really speaks to an underlying desire (though you don't know it) to have per-location subnets with DHCP scopes for each.

To speak to your question specifically re: DHCP: A DHCP "scope" is a range of IP addresses and options that a DHCP server will "hand out". The DHCP server chooses the scope to choose an address based on either the network interface the request is received from (if it's a broadcast request) or the address of the DHCP relay agent (if it's a relayed request).

Some background: Best way to segment traffic, VLAN or subnet?

Windows 2012 R2 DC in main subnet, with DNS and DHCP server, of course

Why the "of course"? The DHCP server doesn't have to be on the DC - in fact, I would argue that if you have other servers available, it shouldn't be running on the DC.

To answer your actual question: You can have the clients themselves perform the dynamic DNS registration which will update the DNS servers with the IP address obtained from the DHCP server running on your ASA.

However, if you have a switch capable of running VLAN's, why not setup the switch to do the DHCP forwarding/relay to your Windows DHCP server so you can manage the scopes from a single mgmt interface for your internal network?