If it has deep sentimental value then you will be worrying about it the whole time it is in coat check, which is not worth dealing with.
But really, things don't get stolen out of coat check very often and I wouldn't expect fur to be stolen more than anything else. So few people wear it, who would buy it from the thief? I also would not worry about someone tossing paint on you or the like: people are as likely to assume it's fake or vintage, as to assume it's real.
There are cold weather situations where fur is defensible. Ethically I am not comfortable with mink or fox for fashion: I eat meat and use leather but nobody is eating those fur-bearing critters after they're killed. And they are farmed in awful conditions. Bison, seal, rabbit, etc is more likely to have lived well and more likely to have been consumed.