No you cannot.
- Forced slinger drops by light weapons' Clutch attacks always happen and cannot be prevented. Later they will simply not drop (exact numbers yet to be known), but this will not improve the weapon's wounding capabilities.
- As you might already know, two consecutive attacks by a light weapon is required to successfully tenderize a body part.*1
- Tenderization and the forced slinger drop happens at once.
However, do note that five claw attacks are equivalent to a heavy weapon attack, and three claw attacks are to a light weapon attack. If you can risk the monster being enraged faster (and thus losing the ability to flinch shot more frequently), you can simply strike thrice with the claw and attack with the weapon to directly tenderize it. Or better, claw thrice, flinch shot, and then attack with the weapon while it is downed.
The rule that "light weapons" require two consecutive Clutch weapon attacks or an attack followed by claw attacks also seem to apply to refreshing the tenderization.
As far as I know, the only exception for this rule is Safi'jiiva, who uses a special rule for tenderization which eschews the "light/heavy weapon" rule and allows every weapon to tenderize with just one strike, but does not drop slinger.
*1: As of Ver.12.01, some peculiar attacks ignore this "light weapon-heavy weapon" rules. Sword and Shield has Clutch Claw Uppercut which does half the tenderization despite not being a weaponskill, while Dual Blades' Spinning Rising Slash which is a followthrough from a special clutch action tenderizes the skin with only one attack despite being a light weapon attack. Aside from these peculiar attacks, however, most points given in this answer are still valid.