Deploy a DHCP server with no IP adress on NIC serving DHCP requests? (Raspberry Pi?)

I have a fringe case. Dont know how to do this. Hope you can help point me at the right direction.

I have a VPN appliance that needs DHCP provided internet connection to connect, and call home. (Kindda like Meraki stuff).
The device needs to be able to pull an IP, DNS and GW info from a DHCP server to operate.

BUT i have a location where the only kind of reliable internet can only be had with a single static IP, and the ISP will not provide DHCP for the connection.
Right now i have made a McGuiver solution, and just plugged in the first available homegrade router i could find at the office. Plugged this in between the ISP connection and the VPN box, just to be able to provide the VPN appliance with a DHCP issued IP and GW + DNS, so it can connect home to our main firewalls.

Since i dont trust this homegrade router to keep running without nursing, i need something better to replace it.

I could spend an arm and a leg to put in a quality router, to take its place. But as the router is absolutely just another route that is entirely unneeded, i would rather find a way hand out those DHCP requests instead.

So now i am getting out of the comfort zone on this.
All DHCP servers i have ever setup (Windows, BSD, Linux, FW applicances all had an IP adress on a interface to be able to set them up as DHCP servers.

Is there any way i can just setup a Rasberry Pi on the same switch to hand out the one IP adress i need, without it having to have an IP on the same network?

The network on the public IP is a /30 (255.255.255.252) network, where my device have one IP, GW have another, one for broadcast, and one to define the net. So there is (to me) no way to place a DCHP server in this adress space.

I have no network training or education. So if all of this sounds just silly/crazy to any of you, i appologize, and just ignore me.