Reweighing the airplane and a supplemental type certificate are not needed to remove the right side control wheel.
If the right side control wheel in a Cessna 182 is an option, then the airplane was certified with only the left side control wheel. Since the C-182 is certified for single pilot operation removing the right side control wheel would be a minor modification.
All optional equipment weight and arm data for that model airplane would be listed on the equipment list that should be part of the aircraft paperwork or would be easily available from Cessna.
Reweighing the airplane is not necessary. The current weight and balance would be amended with the data from the equipment list to find the new weight and balance. All is needed in addition to the updated weight and balance is a logbook entry signed off by a licensed A&P technician stating removal of the control wheel and a new weight and balance has been calculated.
Anytime a modification is done that affects the weight and balance, it would be recalculated using the existing weight and balance. Only if the weight and balance is missing from the aircraft’s paperwork would the airplane have to be reweighed and this new weight and balance sheet would then become part of the aircraft paperwork. This is only for Part 91 airplane’s, the same airplane used in Part 135 air taxi operations must be reweighed every 3 years.
No STC (Supplemental Type Certificate) is necessary to remove the control wheel even if it was not on the optional equipment list unless the intention was to sell this modification to other C-182 owners.
If the owner just wants to remove the right side control wheel if it was certified with both control wheels on his own airplane, a onetime field approval from the local FAA FSDO office is all that is necessary. An FAA Major Repair and Alteration Form 337 would be filled out specifying what was done by the technician and signed by the technician and inspected and signed by an IA
(Inspection Authorization). The FAA inspector could then approve it from just the paperwork or require a field inspection of the modification. This field approval would only be applicable to that particular airplane by serial number and could not be done on another airplane without another FAA Form 337 and FAA approval.