Could the sudden change of speed, with the dropping of the weights, have been the proverbial straw at this point?
Not that this design wasn't inevitably doomed anyway.
Probably not if it wasn't already damaged. Previous passengers on the Titan reported hearing bangs and snapping sounds. In the case of a normal bathyscaphe or submarine constructed of thick steel, those sounds are caused by buckling or flexing of the hull material under stress. Even very minor buckling can sound very loud, and when the vessel surfaces, those stresses are released and the hull material usually goes back to very close what it was like before. But this is not the case with composites. In the Titan, every one of those was the sound of fibres in the hull fracturing, and when composite fibres fracture, they don't go back to the way they were before. There is also no effective way to repair them, so the damage is cumulative.
Titan had visited Titanic four times before this dive, but had done another nine dives to similar depths. Every one of the fractures in the hull fibres on each dive was a new one. From what I understand of the materials engineering involved in building a cylindrical composite pressure vessel, there is no way to guarantee exactly even density and thickness around the whole structure and this will result in weaker areas and stronger areas, and while fractures are likely to happen mostly in the weaker areas, that is not always guaranteed. If several fractures were to accumulate in a small area, this would drastically weaken that area, so yeah, under those circumstances, a sudden acceleration under pressure might lead to a failure. I recall there was a similar problem on Apollo 10 when they were testing the LM.
JayUtah could speak more authoritatively on this than me, but IIRC at one point during the flight, the Grumman head engineer Tom Kelly became concerned when he saw one of the helium tank pressures approaching the upper tolerance limit. He was worried that when the descent engine was fired, the sudden acceleration could cause the pressure inside that tank to spike and lead to it rupturing.
To get a sense of how quickly the Titan implosion happened, the human sensory system requires around 10 milliseconds for the signals from the body's outer sensors to reach the brain, and then the brain takes around 100 milliseconds to "feel" the pain. Titan filled with water and imploded in less than 1 millisecond. The five occupants did not know what hit them, in the truest and most literal sense of the expression.