When a genetically-enhanced soldier escapes from a Military Black Site in New Mexico, his CIA handler struggles to find a way to apprehend him as the soldier leaves a trail of destruction throughout the United States.

The super-soldier is more interesting, however, you have to tell your own story,
The POV character does have to be the protagonist. The most obvious example I can think of is "The Great Gatsby".

The creator of "Breaking Bad" said something regarding framing a "bad guy" as the main character along the lines of: The audience does does not have to agree with their actions, but they do have to empathize with them.

Regarding the genre:
You can introduce the GMO through the CIA agent's POV and occasionally snap to his victim's pov doing fights.

To help with your logline:
POV: super-soldier
Event: escapes black site
Action: Free his brothers
(Story goal: to overthrow the US government)
(Stake: life or death)

This is my attempt, but keep in mind, I know a little about story structure and noting about loglines.

After his escape, a lab-born soldier struggles to free his brethren while his ex-handler follows the trail of bodies that will lead to their execution.

How should I reframe the logline with the GMO as the protagonist/anti-hero? I still think if it's a Military Horror, that looking through the CIA handler's eyes as the GMO is like this super boogeyman, wouldn't that make the GMO more scarier?

To overthrow the US Government and take siege of the White House. His adventure takes him to several Black Sites around the U.S. that contain other GMO soldiers so he goes into them to set them free so he can build an army.