Love the action in this one and the beautiful background. You probably could fix the slight motion blur in the head with Topaz AI Sharpen. As for shutter speed the small birds are a big challenge. Increasing shutter speed helps but, as you said, the higher iso may introduce other problems.
Have you tried adding "fill flash"? Using an on-camera speed light with something like the Better Beamer can really help reduce motion blur. Generally I would keep the flash in manual mode + high speed sync at a very low setting like 1/64th or 1/128th power. This provides a very brief low powered flash (somewhere in the range of 1/15,000 second) which can enhance the effect of freezing motion without over powering the exposure.