The 223 will work to any range he can reliably hit them. 300 is easily done with any bullet weight from 40- 60 grains.

If it were me and 300 was the max range I would load up with a 55 grain Ballistic Tip or a 53 grain V Max and start killing. Sight in at 200 and probably hold "in the hair" to 300.

The last time that bear ate a lawyer he had the runs for 33 days!