Hi, Graham. I don't know much about knife fighting but if you are interested there are some related threads at the bottom of the page and this is some background on the various grips:
The normal grip for a tanto is with the blade cutting edge down
(from what's called a hammer grip in that article). So you handle it like your right hand grip of a katana. Like a katana you grip a tanto from your little finger and your forefinger should be very loose. I was taught shomen uchi and yokomen uchi suburi for a tanto in Japan many years ago.