In “Relativity”, during the scenes before Voyager leaves dry dock, we get a different story as to why Tom Paris would be aboard Voyager. We oversee a conversation Janeway and Admiral Patterson have.

JANEWAY: Catching up with him in the Badlands is going to be tricky, even in a ship as quick and smart as Voyager. I've heard about a pilot who might make the job easier. PATTERSON: Who's that? JANEWAY: Tom Paris. PATTERSON: Admiral Paris's son? Correct me if I'm wrong, but he's serving a sentence for... JANEWAY: He's made some mistakes. But everybody deserves a second chance. PATTERSON: I'll look into it.

When Janeway speaks to Paris at the New Zealand Penal Settlement, Starfleet apparently decided to go a different route:

JANEWAY: You help us find that ship, we help you at your next outmeet review.PARIS: Ah ha. JANEWAY: Officially, you'd be a Starfleet observer during the mission. PARIS: Observer? Oh hell, I'm the best pilot you could have. JANEWAY: You'll be an observer. When it's over, you're cut loose.

That a long way from wanting the best pilot possible to help them in the Badlands. Honestly, it sounds like nothing more than a token gesture had Voyager not gotten stranded. It’s possible he might’ve even been returned to the penal settlement, and given how we see Tom is regarded in “Non Sequitur” it’s even harder to believe he’d be allowed on Voyager as anything more than a prisoner.

STRICKLER: Mister Kim, why did you travel to Marseilles, France this morning? KIM: I went to look up an old friend. STRICKLER: You have poor taste in friends. Thomas Paris is a convicted traitor and a Maquis sympathiser. Now, what did you talk about with him?

Starfleet doesn’t have any love for Tom Paris, and even Admiral Patterson didn’t seem to keen on the idea himself. Even in “Non Sequitur” we find out his leash wasn’t very long after he’s arrested by Odo.

PARIS: What? Are you kidding? I said yes! It sounded better than Starfleet Rehab. Anyway, I got as far as Deep Space Nine, where I got into a bar fight with a Ferengi, and I was thrown into the brig by a very unpleasant shapeshifter. Janeway tried to get me released but my parole was revoked by Starfleet Command. Voyager left without me.

Given what we see in “Caretaker”, I find it almost impossible that Tom started the bar fight. We saw the resentment towards him from Voyager’s first officer and chief medical officer. Perhaps even Ensign Daniel Byrd, who replaced Harry Kim, had heard stories about Tom and resented him as well. He could’ve very likely started the fight, and when the dust settled blamed it on Tom. Who would Starfleet believe? One of their new graduates or someone they’ve already labeled a traitor? That would explain why he seems more bitter towards Starfleet in this reality.

So from every aspect it seems like Starfleet should’ve just kept Tom at the New Zealand Penal Settlement and never let him set foot on Voyager.