Yes
Awarding experience as entire levels instead of keeping track of the experience points is an easy and reasonably popular alternative way for character advancement. The Dungeon Master's Guide has a small section devoted to this variant, "Level Advancement Without XP", on page 261.
You can award levels on a per-session basis, or based on campaign progression. In my own DnD 5e games, we use the latter: whenever the players complete an adventure or a particularly dangerous part of one, they gain a level at the end of the session.
An advantage of this method is that it is possible to fine-tune and pre-plan the experience progression very easily to suit your tastes, whether that means a swift "zero-to-hero" ascension or an extended adventuring on the lower levels. There's also less need for bookkeeping, and the players may feel less pressure to seek out combat and other challenges just for experience. A possible disadvantage is that the progression can feel arbitrary and less objective than XP, and removes the immediate experience reward for succeeding in combat or other challenges.
Ask the player.
First there is nothing wrong with making a mistake, we all do it, you will make more, don't try and hide it. You're the DM you are tracking a lot of stuff mishaps happen.
Just ask that player what they want to do, do it before the session even starts. Pull them aside, explain what happened, (they should be tracking the companions saves and hp anyway, not you) and see what they want to do. They may be satisfied with their companion's glorious death, or they may feel cheated. Find out.
If they want the animal back tell everyone what happened, just like you did for us with what you forgot to do, (that takes care of any sign of favoritism) and then bring it back. If they want to try a different companion, let the results stand, and let them seek a different animal.
Fun is more important than a seamless narrative, if you make a mistake that impacts a players fun, fix it as best you can. If you make a mistake that does not impact anyone's fun, just make a note and try better next time. Heck sometimes a mistake is the best thing that can happen to your story, it's rare but sometimes a mistake can make the game more fun.
Best Answer
This is a case where you need to set ground rules which manage player expectations.
When you allow players to manage their own sheets, they are accepting responsibility for doing it correctly. Giving in to this demand sets the precedent that everybody else has to put up with fixing their mistakes. Mistakes happen in RPGs all the time — would they be happy re-running the scene if you had let them use some ability or power they shouldn't have been able to? Staying firm sets the expectation that they'll carefully manage their own responsibilities. You want to keep the game future-focused, not poring over past mistakes. If something is more than one turn in the past, it's probably best to move on.
Death is a part of the game. Risks need to have rewards AND consequences, or they are meaningless. Your players made the choice to risk it, they took responsibility for their sheets; it might be tough, but they signed up for it. If you give in to their request in this matter, it would seem that you're undermining their own agency.
That being said, I do recommend showing whatever reasonable consideration you can in response. It sucks to lose a character, especially due to a mistake. It doesn't hurt you to cut them some slack.