I agree that the timing seems inconsistent between Hyperion and Endymion; in the prologue of Hyperion, Meina Gladstone specifically says it will take 3 (Hyperion) years to get from Parvati to Hyperion1:
“A FORCE:space task force was immediately dispatched from Parvati to evacuate the Hegemony citizens on Hyperion before the Time Tombs open. Their time-debt will be a little more than three Hyperion years.” Meina Gladstone paused.
In Endymion, Chapter 17, the consul's ship says that transit to Parvati will take 3 months of time debt:
I looked at the girl. “The Ousters? That’s where the old poet said you’d want to go.”
“I changed my mind,” said Aenea. “What’s the nearest inhabited world, Ship?”
“Parvati. One-point-two-eight parsecs. Six and a half days shiptime transit. Three months time-debt.”
It is also true that Aenea mentions in chapter 19 of Endymion that once you reach "quantum velocity", they cannot outrun you:
I caught her gaze. “No, seriously, how could someone be waiting for us at Parvati? They can’t outrun us—we got to quantum velocity first—so the best they can do is translate into Parvati space an hour or two after us.”
“I know,” said Aenea, “but I still think that, somehow, they’ll be waiting. We have to come up with a way this unarmed ship can outrun or outmaneuver a warship.”
It seems likely that Simmons just changed 3 years to 3 months for plot reasons, but if I may speculate for a moment, I can come up with three possible rationalizations:
Given that they mention "quantum velocities", it may be that above c, velocity is quantized, so, for example, you might only be able to travel at speed 2c, 3c, 4c, etc. In this case, it may be that Aenea knows that her ship travels as fast or faster than the fastest speed currently achieved by the Pax, even though that speed is 12x faster than the fastest speed achievable 200 years earlier. By analogy, imagine a world where you could only go either 70 miles per hour on the highway or 500 miles per hour - in that world, a honda civic can't go any faster than a lamborghini, since both can reach 70 but neither can reach 500.
Since only the total travel time is specified, not the breakdown of time at C-plus vs acceleration time, it's possible that most of that 3 years was spent accelerating to "quantum velocities" and then decelerating on the other end. Improvements in inertial compensation (provided by the Ousters for the consul's ship) could allow for much faster acceleration allowing people to much more quickly achieve quantum velocities. That said, given the description in Chapter 24 of Endymion: "He switches to tactical space and hovers above the plane of the ecliptic, watching the red dot spin up to C-plus and disappear from the solar system.", it seems likely that if acceleration / deceleration to quantum velocity was the issue, it certainly isn't anymore.
Hyperion (and I believe Fall of Hyperion) are both in-universe books by the poet Martin Silenus, and it is possible that he is an unreliable narrator for a number of reasons. It may be that Silenus himself just exaggerated the amount of time it takes to get to Hyperion either to emphasize its place as a backwater or to heighten the drama (or because he is an artist and doesn't care about petty details like how long it takes to get between places).
Regarding the St. Anthony, I think that's just confusing wording. He specifically mentions in Chapter 24 that it will take 5 months for the St. Anthony (which enters Parvati space soon after Aenea leaves for Renaissance Vector) to reach Renaissance Vector:
He decides not to wait. It makes little sense—a few hours’ difference in a five-month head start—but he does not have the patience to wait. De Soya orders Raphael to prepare a transponder buoy, and he records the orders for Captain Sati on the Anthony: immediate translation to Renaissance Vector—a ten-day trip for the torchship with the same five-month time-debt that the girl will pay—with readiness for combat immediately upon spinning down to RV space.
So in chapter 26 when he says that the St. Anthony translates hours after the Raphael, he means out of Parvati System not into Renaissance Vector - they left hours apart, but arrived months apart.
 Later in Hyperion, it is mentioned that it also takes 6 "web" years for a Treeship to get to Hyperion and back, so I have ruled out the possibility that a Hyperion year is 12 times shorter than Aenea's conception of a year.