How would any of you TCAS users have reacted in the same situation? Remember, only 2 aircraft within miles (you and the F-16), both visual, the F-16 flying VFR, you flying IFR in VMC conditions.
On the military hardware I flew (Global Express) if visual with traffic that we could see would shortly cause an RA, we would invariably go 'TA only'. We invariably did the same in our instrument pattern when you knew ATC-coordinated traffic was likely to result in an RA which would just spoil everyone's day.
If you're not going to follow an RA then perhaps you should have gone 'TA only' so 'his' TCAS then knows your TCAS is not going to be giving you a profile you're going to ignore! If you're going to keep the TCAS in TA/RA then you have to follow it!