To go begging means to be available for one to take or claim, but in this sentence it is suggesting that laoghaire not be available to take or claim a gaberlunzie badge.
Consider the statement "I didn't bother suggesting that we allow laoghaire to put in for a gaberlunzie badge and not be available for one to take or claim through the parish.". Is the statement "I didn't bother suggesting that we allow laoghaire to put in for a gaberlunzie badge and go begging through the parish." false? Yes/no/unknown