I went through something that was *slightly* similar with my cat, Slipper, about half a decade ago. He was only with us for a weak, but came home to us a bit sickly. We didn't think anything of it given where he came from (PetSmart), but now I wish I would have.
We took him to the vet on day 7, only to find that his kidneys were failing and he was more or less on his deathbed, despite his condition not seeming that bad. They said there was nothing we really could have done - he had been sick even before we got him. But I felt like complete and utter trash for not knowing sooner. I felt that I had let my poor boy suffer for 7 days, all because we didn't think it was that serious.
Granted, we called another vet on day 3 and they said it was fine to wait, but still. I let him down, and that is something that I have, and always will regret.
The only comfort I can offer is that now, at least you know he is suffering no longer. I always felt it was a silly thing to say in situations like this, but we both tried. We can debate all we like whether or not we did "enough", but the fact remains: we tried. Dwelling on the past won't change what happened.. all we can try to do is move forward. To learn. To grow. To try and do better next time.
And despite the fact that you feel you failed him, I know MrTash treasured your time together, and appreciates all you did for him. I hope you, at the very least, are able to take some comfort in this fact.
Proud parent of Scamper (age 12) and Robin (age 4)
At the rainbow bridge: Slippy, Faramir, Gabriel, Lucky, Ammy