Installing on a slightly too narrow frame?

I just bought the 2020 ring doorbell only to realize the doorframe is just barely too narrow to hold the 4 mounting screws securely.

A. I cannot mount it on the siding/wall because a plastic shutter borders the door on both sides.
B. I cannot mount it on the door because it is a steel door.
C. I cannot switch to the pro because I do not have existing wiring.
D. The peephole cam does not have the level of good reviews I’m looking for.
E. I can see some door mounted options, but I’m hoping to avoid them: they don’t look good and more importantly, I don’t want the Hassel of constantly having to take the entire thing down and redo it when the battery has to be charged.

Are there any mounting devices at all available where the screws between the mount and wall are slightly inset from the OOTB mount? I have looked at the angled /narrow mounts, but they don’t specify the width between the mounting screws. From pictures, most appear to be in the same place as the out of the box mount.

How about cut a piece of wood to the size of the back of your Doorbell.
Mount that (with 2 screws?) to the doorframe.
Then mount the Doorbell (with its 4 screws) to the piece of wood.