Those are cases that are a bit different than Titanic. As you point out, in the case of Lusitania both the damage caused to the ship from being torpedoed and the rapidity at which Lusitania foundered really prevented engine orders from being implementable. The case of Britannic is a bit more straight forward than that of Titanic. This is because Britannic was serving in a war zone, and while it was unclear what caused the damage to Britannic Commodore Bartlett immediately knew that Britannic's damage was serious and catastrophic; however, he also knew that Britannic might be saved if he could beach her on the island of Kea, which visible to Britannic's starboard. On Titanic in the immediate aftermath of the collision there was no reason for her crew to suspect Titanic had been as seriously damaged as she was, and the true nature of Titanic's damage was not ascertained for some time post collision.
This is not actually the case. Without going into great detail about the physics and the engineering, moving a ship forward with damage to her bows forces water to enter the hull at a much quicker rate than were the ship to simply stop--backing the ship up in this circumstance would actually have the opposite effect, but that is neither here nor there.
This would have been known to the crew of Titanic and the engineering profession in 1912. Thanks to information from the late community member David G. Brown, I can say with relative certainty that 34 years before the loss of Titanic, naval architect Robert Froude published his experiments that showed concretely that a ship with damage to the bows, when steamed ahead, would hasten its sinking when compared to a ship with similar damage that were left dead in the water.
Everything has a
cause, though. Froude is, of course, correct. Steaming ahead increases the effective water pressure, which with damage on the bows will increase the rate of ingress of water to the hull .... But where does it go? That's the key thing which you have to account for your theory to hold water. If the water can't go anywhere, that increased pressure merely increases the loading on the first internal transverse holding bulkhead. And, in fact,
even if there is flooding aft of that bulkhead, the pressure will still be imposed on it by the ingress of the water from the hole forward of it. Then flooding between Bulkheads 1 and 2 increases loading on Bulkhead 2, and so on and so on until you reach the end of the damaged sector of the hull.
(depending on the exact nature of the damage there may or may not be communication between the watertight compartments which causes cumulative pressure, but the cumulative pressure will be a fraction of the total pressure in any case).
Strained bulkheads were the loss of many a warship that was ordered to maintain speed, like
Lutzow at Jutland, for the needs of military necessity. I believe Captain Smith was cognisant of the risk; even before Froude's calculations it was somewhat known as nautical empirically derived information, just not systemized. But they were lost by the bulkhead
failing.
In this case the evidence for a bulkhead failure is not supported by the modelling of the sinking. That has been covered adequately elsewhere on this forum. Therefore, the only way that running ahead would contribute to her sinking is by forcing the water higher in the watertight compartments subjected to flooding, since the pressure must go somewhere and if the bulkhead doesn't fail then the kinetic energy must be converted into potential energy by forcing the water column higher into the ship.
Therefore, to prove that
Titanic was underway after the sinking, you must do the following:
1. Prove that she sank faster than the condition of static equilibrium through the opened area of the hull would otherwise allow.
Since the existing models track reasonably well with her sinking time assuming a static condition (no pressure spike induced by kinetic energy), you have a problem -- something else must be wrong.
What is it?
The obvious one is that the damaged area is less, and certainly, I would be very interested if you have evidence for that.