Could mesh from an inguinal hernia surgery cause chronic pain at the base of the penis/scrotum area? If so, what can be done? (other than painkillers)

Nerve entrapment. It could if nerves were entrapped in the repair. This may also be effected by the use of permanent versus absorbable sutures. Hypersensitivity is not unusual for 2-3 weeks, subsiding thereafter. If yours has persisted longer than that, notify your surgeon.