I'm so sorry you're going through this, doodlebug.
When you furbuddy is dying, your choices are: A) both of you suffer, or B) you grit your teeth and do what you know will make you suffer but will ease your friend's misery.
This is tough for me to say and tough for you to do, but you are going to feel bad either way. Grief and/or guilt. But your buddy will be out of pain if you can be strong.
Good luck to you, doodlebug! I've been there and feel your struggle.
Secondary Progressive Multiple Sclerosis (or maybe RR), diagnosed '00
Everything will be alright in the end. If it's not alright, it's not yet the end.