Your plane is legit. Don't worry about that. When the belly putter is anchored in the belly it is still the shoulders that are doing the work. So, you are tracing a straight line on the ground with your forefinger which moves the putter head in an arc.
As far as the straight back and through business. Staight back and through is possible, even wanted, if your upper spine is horizontal to the ground. Most, if not all, players do not have this accomodation at address but it is possible. So the arc type stroke will fit you best unless you change your posture.
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