Is there a best way to remove this (small, flesh-colored, raised) mole from my nose? (photos)

The fleshy bump on your nose can be removed easily. In my office the area is numbed with a local and then the raised portion is shaved off even with the skin around it. The base is then lightly cauterized with an electric needle. You get a small scab the size of the lesion which heals in about a week with no pigmentation and little to no visible scarring.  We always send the removed portion to the lab because every so often the bump is an early skin cancer and that can't be determined without sending it. Find a reputable dermatologist and it should go as I described above.