I would definitely stick to the Pardiso solver and automatic incrementation, but you can play with other settings including meshing parameters. Maybe try switching to 3D now. Then you can also use rigid body constraints and other types of contact (e.g. Mortar) without issues common for 2D elements: Known CalculiX limitations
Of course, modifying the geometry (different relative positions and fillet radii) may also help. Maybe CalculiX won’t exhibit that snagging or mesh distortions if you do that.
Yes, as I said initially, that’s one kind of problems where CalculiX struggles a lot compared to e.g. Abaqus. It may often work eventually, but after many attempts. It’s particularly sensitive to mesh and contact stiffness. See the snap-fit example on the CalculiX forum. Or this one: Insertion and withdrawal force calculation with plastic deformation
Actually, even before your post, I was going to write on the CalculiX forum (necessary before submitting a GitHub issue) because it would be good to discuss it with the devs. Maybe they can suggest or improve something when it comes to handling such cases. Can you export your .inp file (with a fine mesh) and share it on the CalculiX forum (via some hosting website) ? I have some other examples too, but I would appreciate if you could share yours there.
Alternatively, you could also try OpenRadioss instead of CalculiX.