I would not recommend doing it that way. We don't know exactly what's going on with your bird, it may not be digesting and absorbing normally, which could make it ineffective or much, much too slow, which could increase suffering. When using pain killers they are normally given intravenously in large enough amounts to be nearly instantaneous. When euthanizing it should be as quick and painless for the bird as possible. Cervical dislocation is often the best and easiest method for home euthanasia, it's easiest for the bird, but is not hands off for the human. The following article might be helpful for you, I do it this way for many of the same reasons as the articles author sites. This is unfortunately a needed skill if you are going to keep chickens, unless you have a vet that will do for you at a reasonable cost that you can handle.
Even then, sometimes circumstances dictate a faster timeline, to be kind to the animal, if an appointment cannot be made pretty quickly.
https://www.muranochickenfarm.com/2017/06/the-easiest-way-to-cull-chicken.html
I understand how you feel, honestly. It is not something I enjoy either, but it's necessary, and our responsibility as animal owners. And despite the emotions involved, it is very often the greatest kindness you can do for them. In truth, the anticipation of doing it is often much worse than the actual doing of it is. With the method above, you can do it, and walk away for a few minutes to let your emotions settle, let yourself breathe, and then come back.
