It is unknown which grandfather is named Lawrence the one on the Laffer side of the family or the one on the Lovage side of the family. If the former, it may actually be Larry Laffer's father (although some rumors suggest that his father is Al Lowe).
The Laffer side of the family naming their decendents Larry or Lawrence appears to be a bit of a tradition going back possibly to the 1800's (Dwight L. Laffer) assuming L. stands for Larry or Lawrence.
In the Reloaded universe Larry's father may still be alive. He mentions him when examining a Wurlitzer. He describes his dad as being cold and unyielding.
His father gave him this advice: "Full moon by day, girls stay away. Full moon by night, women take flight."