ER16 is definitely too small to drive that large of a tap. I used to have trouble making an ER16 drive a 3/8 UNC tap even with a floating tapping head.
I always apply tapping fluid to my taps because I get mad when a tap chips early in its life. Coolant is not a tapping fluid, IMO.
I agree with Geof about repeat tapping, although I would probably only use one repeat because I use tapping fluid.
Best way to figure out what works best: hand tap a hole full depth using coolant for a lube. Hand tap another hole full depth with tapping fluid. Hand tap another hole with a retraction half way down. Feel the difference.
Haas makes some kind of a lubricant squirter that will shoot a shot of lube at the tap. I presume such a thing was invented because there is a need for it.
If feasible, program all your tapping near the end of the program. This way you are going to be there soon to change the part anyways, so you spend a couple of minutes with a brush (door interlock disabled, but still keep your face behind the glass because when a tap shatters it can explode like a bullet). Also blow coolant off the part before tapping to avoid blowing off the spent lube into the coolant. There may be some lubes that are coolant compatible and won't create an oil slick.
First you get good, then you get fast. Then grouchiness sets in.
(Note: The opinions expressed in this post are my own and are not necessarily those of CNCzone and its management)