So basically, you want to call Fox, Bobcat and Coyote. And If they show up and you can see them, you want to be able to put them down right there and you want no fur damage.
In my opinion: There is not a cartridge in existence. As a caller, the closer you can call them in the smaller the cartridge has to be. However non listed above will save fur every time at ranges under 100yds.
1) Rounds traveling over 3000FPS at close range tear fur, more so on something delicate as a Fox or Bobcat.
2) FMJ bullets may pass thru, and leave a small exit if they don't hit bone, but animals have the tendency to run.
3) Soft nose bullets will mushroom, more so if they hit bone. They will usually leave a hole for the seamstress.
3) Ballistic tip ammo can come apart on impact, (splash) or pass thru failing to preform. Now if the distance is right starting at this range(xxxx) to range (xxxx) and the bullet enters the chest without hitting the shoulder and the bullet preforms, it may not pass thru blowing a hole that you have to sew.
4) Hollow points open when they start to meet resistance (Flesh), there is no polymer tip that needs to be pushed back into it for the bullet to expand.
Bullet manufactures design different bullets types to preform differently. So the only thing we can do is manipulate the bullet by speed and place our shots.
No Magic bullet: I have always tested every bullet for the calibers that I have used and stopped when I found one that preforms the best for my needs.
All 3 calibers listed above will kill Fox, Bobcat and Coyote, if kept within range of what the shooter abilities are.
Coyotes are in a class of their own, Their hide is heavier and their bodies are larger. They can
absorb a hit a little bit more. Regardless of caliber, Use the right bullet, keep it in range, place it where it needs to go and you will drop the majority right there with little to no fur damage.
Read all you can about the caliber, the bullets and then preform test Yourself. In the right hands the 223, 204, 22-250 are deadly deer killers thou people will say other wise.