John, I know people who use the same grease they use to grease there trucks and have had no problems. The reason most of us are useing the thicker or heavier grease with the beeswax is to help keep it from melting and running out the end of the barrel on hot days. I could be wrong but it seems to me that it's also much , much cheaper than good litium grease.
Give it a try and see how you like and go from there.