Rendering the third person model in first person also means rendering the cape (since it is part of the third person model).

This becomes a problem because the cape can sometimes clip into the camera.
This currently happens when sprint-swimming and looking down.

The problem becomes even worse when using cape physics from physics mod; you need not even look down, but straight ahead to see the problem:

Therefore, an option to prevent the cape from rendering whilst in "first person" would solve this issue at the user's discretion, similar to the approach towards arrows/stings being a togglable option.

That doesn't seem to be a normal cape. Never seen heard of the cape getting into the way, since then also the torso would be in the way.

That doesn't seem to be a normal cape. Never seen heard of the cape getting into the way, since then also the torso would be in the way.

Well, after further investigation that's mostly true. Using the mod version 2.2.0 on Fabric 1.19.2 alone, I could still sometimes get the cape to clip into the camera while sprint-swimming and looking down, but it wasn't nearly as bad. Any FOV at Normal or below should not see the problem, but I think the majority of players set their FOV higher than Normal.

It seems like the Capes mod is what's making the problem so bad (in the first two images, the third image is the behavior alongside the Physics mod's Pro Version capes).

It's obvious why the Physics mod causes this problem, but I'm not sure why the Capes mod makes it so bad. Capes Mod does not appear to change the cape's behavior or position on the model; only the texture. I'm guessing your mod already has some near-field culling code that the vanilla cape adheres to, but the Capes mod overwrites the vanilla cape and thus doesn't adhere to the culling?

That all sounds more like an issue with the swimming animation, that the offsets are not good enough for some angles/fov. Nothing to do with the Cape.