It's not just that this is a small ship - this is a small ship that has suffered casualties and then taken aboard the remnants of another ship's crew and its chain of command. It would only be expected that the ranks be out of synch with the positions and responsibilities.
That is, unless Janeway engaged in promoting people into rank corresponding to the position. But the only promotion we ever see, that of Tuvok's from Lieutenant to LCDR (or LT CMDR as Starfleet abbreviates it), seems to be of the usual "he did the song and dance for sufficiently many years" sort - not done for organizational reasons or as a reward for heroics.
And if Tuvok really was LCDR in the first season, then there were zero promotions during the show, just demotions and then eventually a pardon for those. Perhaps Janeway thought she had no mandate to promote anybody when there wasn't an official Starfleet letter of approval available?
Tuvok's first year can be treated as either a costuming error or a dialogue error. In dialogue, he's Lieutenant, not merely in the informal "I am Janeway's trusted right-hand Vulcan" sense, but in the "I am Lieutenant Tuvok, Service Number 47LOL227" sense. In costuming, he remains Lieutenant Commander until "Cathexis" where he suddenly begins to wobble... Neither can be considered an "in-universe mistake", as this very solemn character would have censored himself soon enough even if his fellow crew did and said nothing. But if we choose to ignore one of the two mistakes, we could just as well ignore the dialogue and establish Janeway as a CO who never promotes anybody. At least that'd make Harry Kim feel a bit better.
As for the rickety barge thing, the casualty list from "Imperfection" populates the Voyager with at least one Commander and several Lieutenant Commanders, so Lt Tuvok might have been a long way down the chain of command originally. Perhaps the brand new supership was prestigious beyond her real operational or technological value? Or perhaps there were plenty of "observers" aboard during the first mission, and even Janeway would soon have handed the ship over to a more fitting, that is, more junior commanding officer?
Timo Saloniemi