When you dated your husband, did he express his bigotry? In other words, has he always been this way?
In my personal opinion, you do not have to listen to his bigotry. Can you tell him that it is inappropriate and unacceptable to you. To please refrain while he is in your presence? Also- would it make a marked difference if he read what the 1st Presidency says about it? If so, get on lds.org; Gospel Library and hunt down all the talks and lessons regarding such.
I can pretty much tell who my husband has been working with, by the language he uses once he comes home from work, and the crude tasteless jokes.
Fortunately for me, I can look him straight in the face and tell him to clean his language up. Then I immediately ask him to pray and banish the adversary from our home.
You can do the praying to banish the adversary from your home- you have been given that power/gift at your confirmation when you received the gift of the Holy Ghost. Use it righteously.
Going to your Bishop regarding this is NOT going behind your husbands back. You have every right to council with your Bishop and to receive guidance from him. You also should go to your Bishopric and ask for blessings.
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Looking back on things, the view always improves.
Impollutable Pogo (1970)
I'll tell you, son, the minority got us out-numbered! ~ Congersman Frog
(Walt Kelly's Pogoism's)