Personally, I wouldn't use Cody if it gets such a terrible reaction from two close relatives.
Camryn/Cameron would be perfect for you.
Kendall - doesn't start with a C but has the same sound as Carson (Kendall was originally used on males)
I would chose something different. The scars of being cheated on never really go away, they're something that stay with you, even if you have moved on. Now, it's your choice whether to knowingly do something that would hurt your SIL, or try to find something else you love.
Some other "gender-bending" C options:
Casey-More unisex than truely gender-bending, and also maybe too similar to Carson.
Cameron-Same as Casey, more unisex.
I agree with your IL's. I wouldn't use a name that has that kind of baggage in the family, even if it's not your baggage.
What about Colbie like Colbie Caillat?
What about Camden or Cameron? They are a little close to Carson, though. It's difficult to find another C-name that is unisex and isn't too similar to Charleigh and Carson. Is Cadence to girly for you? The only other one I can think of is Chyler (pronounced Ky-ler).
I agree, using the name isn't worth upsetting the in-laws if they feel so strongly about it. I love the name Juliet, but would never be able to use it for the same reason.
I really like the suggestion of using Cameron. I've always liked that name and to me it's very unisex.
I rarely think that you should change your choice due to the objections of outside people, but this one I'd just change to be polite. There are plenty of cool names out there, you can find something different enough. You wouldn't want negative feelings to surround your child's name... in the end, that's really not a fair burden to put on your child. And it would be a bit insensitive to say "Just buck up and get over it..." She went through h***... gotta respect that.
Gender bending is NMS at all, but what about Clancy, Corbin, Caelum (Kai lum, Latin for sky), Cai, Cian, Ciaran (this one could work well, I think), Caillou, Casey, Cavan, Conley, Casper, Cruz, Cameron, or Camden?
Yeah, I definitely wouldn't use it. It has a lot of baggage and I wouldn't want to do that to my in-laws.
I'm suggesting Corbie. My sister just named her daughter that and it's adorable and very similar to Cody.