How do I feed a monster drugged meat?

This is mainly because monsters are more wary of the surroundings in World, including even Deviljho. To drug the monster, it must be:

• exhausted (obviously),
• carnivore (do not expect that Diablos or Kirin to be drugged), and
• not engaged.

The third part-in which the subject must not be engaged-leads to a trouble because monsters will prioritize in shrugging you or other invading monsters off. However, since the monster will momentarily stop before actually fleeing, and even try to fight back moreso in Iceborne, you can use those moments to put the meat and then hide the offenders' trace and make it focus on the meat instead. To hide yourself, you could hide in a bush, use a Ghillie Mantle, a smoke bomb, or just run away. Fending off other monsters is much more difficult since there is a chance for the monster to flee before the dung pod even works.

Also, (Ebony) Odogaron has a special gimmick in which it eats meat to get frenzied even without being exhausted. Normally it reaches for the meat it always carries, but if there is a drugged meat, sometimes Odogaron will try that instead.

Personally, though, since you mentioned recuperating such as sharpening, if you have access to such hiding methods, I would rather just use them directly for this purpose. For sleep-bombing, your Palico, Sleep ammo/coating or sleep weaponry seem to be more realistic, and for poisoning, well, there is a poison smoke bomb.