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yes- cores of the crater would make or break the impact hypothesis very easily- since the site is under water and has in fact had further sediments deposited onto it, the original mineralogy of the sea-bottom crater should be intact. Even if it was destroyed, then there should still be distinctive 'shocked' mineral crystals in the bedrock underlaying the crater.(this is one of the nicer features of large impacts- they leave behind evidence in the nearby bedrock that wasn't macroscopically deformed, meaning that even if the entire crater structure (surface and subsurface) was erased there could still be evidence of the impact. of course, without the crater structures no one would look for these signs)
the problems here are that the geometry of the basin suggest an impact (as does the 'nipple' structure), but such extensive and symmetric concentric rings haven't been observed except on icy bodies elsewhere in the solar system- where the topmost layer is the most rigid and the ones beneath are more plastic or even fluid- it should be the opposite on earth. Of course no salt recession basin has ever been seen with so perfect geometry, but if it's true that the area is littered with similar features and that they correspond with thinning in the salt layer the salt theory would be a lot stronger. The fact is that the multitude of rings actually is a bit problematic for both hypotheses- it could theoretically be possible for both, but has never been observed under the circumstances for either.