I'm reading between the lines and there seem to be a few tightly related questions here about running services while conserving power:
Can I run applications or services like pihole when the system is suspended?
Unfortunately not. When a system is suspended, all actions are halted. See: What is Suspend?
Can I run Ubuntu Server on my home network and will that save power compared to Ubuntu Desktop?
Yes. And Yes.
You can install Ubuntu Server on your device and you can run services like Pi-hole on your home network. Actually, Pi-hole is designed to run on home and other local and private networks.
Ubuntu Server will use less power than Ubuntu Desktop because desktop environments (DEs) are fairly heavy on resources. It takes a lot of software running simultaneously and harmoniously to power a DE. Less work for your CPU, hard drive, and other resources will result in lower power consumption.
However, you will not have the same level of power savings that you would get from suspending the system. The server needs to have processes running in order to be available to interact with the other devices on your network.
You also mentioned installing a DE on top of Ubuntu Server to save power. Unfortunately, there are probably not any significant power savings to be had if you install Ubuntu Server + a DE compared to Ubuntu Desktop with the same DE. Arguably, there is some variation in power consumption among different flavours and DEs, but not having a desktop at all is going to save the most power.
Here's a few additional notes on Pi-Hole. If you're only really planning on running Pi-hole on this particular hardware, you don't really need a DE. It will install a web based UI that you can control from a browser on other computers on your local network. Pi-Hole also has excellent internal documentation and they have an active support community.