A couple points -
First - shooting it down over the US has a risk of it hitting something important on the ground. House, child, school, whatever. You can’t control the trajectory after hitting it and there is a chance of fragmentation, yielding more objects hitting the ground. Sure, odds over Montana are low, but they are not zero. And they need to be zero in this case.
So, shooting it down over water was the right thing to do.
Next - a fighter gun was never going to work. A fighter gun has a practical rebate of about a mile. Beyond that, bullet dispersion and projectile deceleration reduce the gun to minimally effective.
You simply can’t get an airplane with gun to a mile away from something at 90,000-100,000 feet. Even the “zoom climb”, if it worked, leaves the fighter as a ballistic object - without the fine control needed to aim a gun at a target a mile away. You aim the gun by flying the airplane. A few feet of aiming correction at a mile is about 1/100th of a degree in pitch or yaw. The airplane has to be responsive and flying for that to work - not on a ballistic trajectory.
So - a missile is the best option. But which one?
It wasn’t a sidewinder. The AIM-9X tracks on heat. The ballon may, or may not, have a heat signature. Likely not, as it has cooled to ambient temperature of about -40 degrees. It’s warmer than the cold sky above it, but not by much. Further, the range on the AIM-9 isn’t that far. The fighter has to get the missile close enough, and with enough velocity at launch, to intercept. Low chance of success with an AIM-9.
The AIM-120 AMRAAM is my guess. Long range. Radar guided. The fighter would be able to see the balloon on radar, allowing a lock and a missile launch.
The launch window would be brief.
Because shooting too early means the missile runs out of energy before intercept. The rocket motor has a small, finite, burn time, after which the missile decelerates. The missile needs to be going fast enough for the fins to be able to adjust its flight path so that it intercepts.
Shooting too late requires the missile to make a steep climb, where the air is very thin. It is limited in maneuverability in that thin air, and a late shot won’t be able to make the intercept because of the missile’s limited ability to turn at that altitude.
At the speed of the fighter, over Mach 1 to be up above 60,000 feet, the launch window, between too early and too late, will be brief. It might require a pitch up to give the missile an initial vector in the right direction.
So, Raptor, AMRAAM, over the ocean.