Those are ECS24200, which is currently standard with the US bundle. What’s not apparent in the photo is that when installed that way, they extend about 1.1" from the face of the cable, so you need that much clearance to install the panel cover. It doesn’t look to me that you have that much clearance, but possibly you do.
Regardless of whether you went with the clamps or the ECS24200, you would need at least 1/2" clearance between the cables for up to 2". I think that’s doable, but can’t recommend that you manipulate those live cables to make it happen. I have to recommend you get an electrician to possibly slit the gray cable jacket as far as possible, and persuade the two conductors apart with a non-metallic tool like a wooden wedge. It’s probably not possible to shut down the feeds unless you have an outdoor disconnect, but at least shutting down the mains would insure there is no current through the connectors as they are disturbed. An electrician may also see a way to feed more cable into the box, and could also disconnect the neutral to install a clamp while the main is off.