Ignored By Bil Since Many Many Years

OP break the ice with BIL.
You don't have to develop a bond with him but just a hi bye how are you kind of talk should be fine.

Since you both aren't directly involved in any altercation , you don't owe an apology to him and I'm assuming whatever happened is because in your bad marriage days your hubby would have shared things with his brother. Now that everything is back to normal with husband I don't see why BIL is still angry on you. He may not be angry but kind of in an egoistic situation where he is not able to talk to you.

I think you should have a word with your husband and tell him to advice his Lil bro to not have any grudges on you . If you have done this then pls ignore.

Next time when a situation comes either at your house or his their house break the ice with a simple " how is your work " kind of questions.

You being elder(in terms of age is my assumption, but relationship wise you are elder to him) to him I feel there's no harm in you breaking the ice. This would do good to you on a long run as this situation

is avoided.

 

He has a tough life ahead is all I can say.
Since you have attempted from your end to talk to him, I don't see any reason for you to worry about anything going forward . If MIL objects to certain things I would say ignore their attitude royally.

This is important and keep reminding yourself this.

And wanted to add that no he's not worth your time.

 

Apart from this tension between you and BIL, how is his relationship with your husband and daughter? If he is a good brother and uncle, I would rather not strain those beautiful relations. So maybe sometimes go for a nice walk or shopping or temple or parlour and let them have a nice time by themselves.
He is unmarried so it is likely he is not mature enough to understand the intricacies of married relationships. Could be that something you said or did to his beloved brother and/parents that he can't get over even though they have. Give him time he will come around. Also retrospect whether you did something that might have hurt him so much. See if you can see your fault and tell him why and how you came to that stage. Say sorry and promise him you would love to keep the cordial relationships that have been established since. If he is such protective person to hold grudge for years, he also is capable of loving your daughter so much.
On the flip side if you are sure this is unfounded grudge, just ignore him and stop even thinking about any of these issues. Focus on your life and leave him be.

 

Read the above excerpts from your post.
A girl-child should stay in her familiar grounds. Never alone with weird, unmarried uncles, that her mother has opinions about.
Swallow your distaste for your BIL, and watch your child for her own safety.
If you choose safety for your child, you'd also achieve a few other benefits you'd already listed.
[ since you had asked for "any input", I had given you what I think]

 

Some relationships are hard to ignore. This is one such. I think your husband should have a matured talk with his bro and ask him to put the past behind and treat you with basic civility and respect. You should ask your husband to do this. If he doesn't, may be you should do that in your husband's presence. Have an open conversation with your BIL about his current behaviour for some past reasons. Provided, you think, relationship with your BIL is worth this move.

Without addressing the core conflicts where this un-forgiveness is stemming up from, trying to say some random formal stuff when meeting each other and expecting him to also do the same will do no good. If you think he lacks that maturity and courage to face the issues in an open conversation, then you should learn to ignore too and carry not hate.