I am not sure I’m smart enough to follow your argument here but I get the impression that you are talking about the increasing expansion rate of the universe (for instance due to dark energy) whereas I am talking about Hubble’s law (showing that the velocity at which distant galaxies appear to move away from us is proportional to their distance). Again:

If the velocity at which distant galaxies appear to move away from us is gradually increasing the further away they are, that would be a proof for the Big Bang statement that space itself is expanding: the further away to start with, the more space there is in between that can expand.

If instead we had a Big Blast INSIDE an existing space (ordinary explosion), at least I would not expect any such effect.

The fact that the whole universe is also increasing its expansion rate is a subject for another show, to quote Pamela and Fraser.