I have tried providing power and looking at the board through a thermal camera, but the resolution is not powerful enough to determine which capacitor is shorted.
I'm assuming you see no thermal difference at all - it will be visible if enough current is injected.
Seriously, place a voltage-limited supply on it and crank up the current slowly while observing in visual and infrared. If the short is a very low resistance (milliohms), current may have to reach an amp or more before anything becomes visible. Do this very slowly; the board has significant thermal mass and a hot trace in layer 9 could burn out before even being seen if rushed. Try 0.25A for 15 min, 0.5A, 0.75A, etc.
Is there any other way the short can be determined? I originally thought to check continuity.
Continuity is not precise enough; a standard resistance measurement may not be either. Some handheld (and most bench) multimeters have a "high resolution" mode, usually giving an extra digit or two. In either case you'll need the ability to discern differences of 0.001Ω at least. If yours can do that, then you can probe (unpowered) across each cap for resistance - the lowest is suspect; remove that one and try the board again.
Isopropyl alcohol is another trick; put some in a spray-bottle and lightly mist the suspect area - the warmer areas evaporate first. Of course remember this is flammable.
And another device is the Polar ToneOhm which has been around a long time and can physically direct you the the lowest-impedance area. Of course, this is a lot of money for a specialist device which could get very little use.
While it could be a cap, it may also be the FPGA, or even "no-clean" flux issues. If it were mine, I'd soak it overnight with mild agitation and check in the morning, just to rule that out.
If all this is still inconclusive, remove the FPGA and check again. At least this way you'll know for sure where the problem lies.