I use a 22-250 quit a bit, but I don't think it quit gets to 4K with the 55s I use... probably pretty close. I was thinking more a long the lines of a 50-58 grain bullet from a 243.
To be honest, it's hard to rock them to sleep. They typically need a little help via medication. A 50 grain sleeping pill administered at 4k FPS always works.
The ones mentioned above will do what you need...but they're over kill. They have features that will serve no purpose for you...in other words, your going to pay for "useless" features.
Get yourself a bare bones detector that can be manually ground balanced, and a large coil. I recommend a...