This event happened in the past, the 2360’s I think with the Enterprise having less advanced security features and procedures. Surely Starfleet security improved how command codes are distributed and shared in the preceding time before this weeks episode. Is Dal locked out of *full* access of the ship with only low level command codes or does he have full command access thus full control of the Protostar as a standard Captain of a Starfleet vessel would? Does the living construct have more control of the ship than Dal, thus making this device the true captain of the ship? We will never know unless Dal takes the Protostar somewhere remote and away from civilisation (in particular a way from the Romulan sun) and tries to initiate auto destruct in order to save the Federation and the Romulans. Dal does not need to die in the explosion he can use an escape pod or get beamed out just in time. The living construct can not just stay on the ship forever, someone needs to get rid of it and I think that Dal should as he is the Captain. There are plenty of places in the galaxy where he could take the ship if the explosion was going to be big…. Just out of the way of any planets with life on. The explosion would probably only be big though if it was in a sun like with a trilithium device.
I don't think that Data taking over the Galaxy class back in 2360-ies is a bad example... plus he's an Android capable of mimicking other people's voices with absolute precision - it really wouldn't have been difficult for someone like him to commandeer a really secure vessel.
Sure, encryption would have improved in 20 years, but so would have measures for breaking through them (probably).
Also, the Vau'N'AKat who initially hacked/corrupted the Protostar are from about half a century in the future (when measuring from 2383). They were familiar with more advanced SF ships and technology and in their interactions with UFP since then probably managed to advance their own to get extremely close to SF's (much like Humanity managed to almost catch up to nearly other species in the AQ with the NX-01 and having only 90 years since first contact - the Vau'N'AKat were ahead of Humanity on the technological curve during their first contact, so its reasonable to think that they would be able to get similar ideas like humanity did from just observing SF tech).
Anyway, in regards to the level of access Dal has... I suppose its possible that Dal has full access to the ship like any other captain would have.. but its also equally possible his access is somewhat limited by default... or, limited in the sense of his own lack of knowledge of inner workings of a SF ship and its full capabilities).
As for whether the living construct has more access than Dal... it did not. At least not initially.
Now that Dal gave his command codes to holo Janeway, the living construct is able to operate the ship independently through her.
Gwyn also gave holo-Janeway upgrades in episode 9 - aka the ability to have solid form and combat opponents, and she was protected against her father's tampering (since she is fluent in Vau'N'AKat and other languages, advanced tech from the future, and other ship operations, it stands to reason she would be able to give holo-Janeway these protections).
I suspect the construct by itself didn't have the means to do what it was programmed to do seeing how the flesh and blood Vau'N'AKat wanted to operate the ship and bring it to SF to destroy it (at least that was the original intent).
But as we saw, Chakotay and his crew managed to break from captivity and reprogrammed the Protostar to leave back through the anomaly on its own.
Its possible that Chakotay managed to do something to the ship when he programmed it to leave Solum without any biological input (only on automation) and this may have corrupted Vau'N'AKat code to a degree (or forced it to adapt to infect holo Janeway) and subsequently it left the door open for the kids to gain access to the ship and reprogram command level access.