I think that it would be safe to assume that a meson gun with a range of 500,000 kilometers is a spinal mount.
So, let me get this correct. Your sensors report a target at a range of 500,000 kilometers from you, or 1.67 light seconds. You now know where the target was 1.67 seconds ago. Presumably, the target is moving at a reasonably high speed, on the order of many kilometers per second. You then align your ship so precisely that you can hit a moving target of say Azhanti High Lightning dimensions, 405m long by 61.2m wide by 36.4m high, at a distance of 500,000 kilometers with a meson particle beam with the meson particles timed to decay inside of the target hull. According to T5.0.9, page 367, meson particles travel at "near light-speed". I will assume light speed to simplify things. Your particles have to be timed to decay with an accuracy of at least one-ten millionth of a second in order to order to decay within a variance of 30 meters. The beam arrives at the target range 3.34 seconds AFTER your last firing position data. In the mean time, your target has traveled for 3.34 seconds at a velocity of maybe 50 kilometers per second. You know precisely where your target is going to be after it has travelled 167 kilometers from your last position data, and you know to within 50 of so meters precisely what the range to the target is going to be, at a distance of 500,000 kilometers.
Note, your target can alter course in three dimensions, including decelerating. If it changes course by one degree, at the end of 3.34 seconds, it will have changed the straight line projects course by a matter of 2.91 kilometers.
I invite all who read this to think long and hard about the likelihood of hitting the target with your meson bean.