B737 maintenance technician reported questioning the need for inspection buy off of an aileron control tab weight prior to installing it on aircraft.

Narrative

During the discussion for an ok to install a right aileron control tab the question about its weight was brought up if it was sold to inspection. I looked up the different non routine that repainted the tab and a weight was made but was not bought off as Required Inspection Item (RII). I stated this on the non-routine and then the inspector rejected the same non routine. Then management and quality control supervisor said the re weight of the part is not RII. The quality control supervisor was Person A and the supervisor was Person B. I disagree with this according to how I read the General Maintenance Manual XX-XX-XX. I believe the weight change or verification of a flight control or tab is adjustment and rig of flight control per ATA XX . The same airplane had two elevator tabs being weight and balance with inspection task. So; this confuses me why the difference. I was asked that someone else get the ok to install in which they did. I then was ordered to install the tab which I did. I continued with the procedure using the weight that was on non-routine number XXXXXXXXX. The non-routine I was working to install it is XXXXXXXX. The number of added weights to the aileron depends on this weight. Ref aircraft maintenance manual XX-XX-XX-XXX-XXX for install and Structural Repair Manual XX-XX-XX for weight needed.I am looking for more clarification. I hopefully am wrong but if the weight depends on changing the aileron counterbalance I feel this number should be verified.