be very careful, i would worry that is a sick one and a risk to any one.
if a coyote did that to me I would give the good old buck shot, (no different if a bear went after my bees)
i don't know where you get the idea they only take what they can eat lol. they will kill and take down calves and only eat a little or just some guts then leave it to kill another animal.
i have a crossbow, i would highly recommend not using a crossbow for coyote defense, or control. keep in mind one bad shot and now you have a wounded coyote, which is already shown to be a risk.
if you have to use a crossbow go with a good hunting broad head.