I also think it is quite impossible to quantify. There must be big differences in the levies of troops in the different wars which took place in the 3rd century (esp. against Pyrrhus, and then against the Carthaginians in the 1st and even more in the 2nd Punic War), and the possibilities for Rome to recruit slingers at all.
As the Balearic Islands were in Carthaginian hands until the end of the century, Rome probably had no access to the mercenaries there, but maybe to Rhodian or Cretan slingers, apart from the slinging auxilia they could raise in Italy itself.
I would say that they generally used a smaller number of slingers than in later centuries, but that is just a guess.