Maybe the influence was directly from the sound of people 'making a racket' - a variation of the phrase is first used in the sixteenth century, and appears in Shakespeare's Henry IV Part 2 'But that the Tennis court keeper knows better than I, for it is a low eb of linnen with thee when thou keepest not racket there.' 'Keeping a racket' would now be 'making a racket', which dates from 1644. The OED gives the source of 'racket', first noted from 1565, as 'perhaps imitative'; 'racket' the game was documented from c1425, so maybe the sound being imitated was the sound of the ball game with paddles played in an indoor court.
So did people making a racket around Racket Court inspire the debtors directly to play rackets against the wall of the Fleet Prison? It must have been a penetrating noise to make it through the general hubbub of early-nineteenth century London over such a high wall.