My opinion is that you should use the pump. If you doubt the pump, take it apart and check it. It is a mechanical piston/barrel pump that is very reliable. There is another post here where someone asked a similar question. In reply to that question I posted some photos of the inside of the pump.
If you take the pump apart, be sure to remove the input shaft before taking the pump barrel out. The pump barrel has a very close tolerance within the body of the pump and is coated to make the outside very slippery. If you take the barrel out of the pump before removing the input shaft the worm gear on the input shaft will score the coating on the barrel and damage that part of the pump.
Once you have the barrel out of the pump you can easily see if it is in tact or not. There is very little that can go wrong inside the pump. But, if grit has entered the pump it can damage the barrel and allow oil to leak past and too much oil will enter the cylinders.
Except for a break in the input shaft, it is my opinion that a bad pump will put in too much oil, not too little.