Welcome to PC.
Does your SIL have enough support for her illness? If not, maybe kindly encourage her to see a therapist and/or join a support group.
Honestly, and I mean this with respect, it sounds like the family dynamics aren't as healthy as they could be. This is not all on your SIL.
I'm not sure why you'd feel your walking away while laughing would help in any way, is respectful of another person, or is excusable. Sure, it's probably better than yelling, yet it's still a pretty low blow. This type of a response is going to build anger/hostility, of course.
There are no good excuses for anyone using anyone poorly, no matter the diagnosis or lack thereof.
Can your family sit down as a family and talk this over, while including your SIL? Can your family draw up, discuss, propose some healthy boundaries for all to respect in family exchanges?
You are wise enough to look for some help here. Maybe you can take the lead in turning this around in your family? Maybe you can propose some changes and set an example?
Have you heard of NAMI? They are all over the U.S. and they work with families, offer educational series, offer support groups, etc. I hope they are offering groups in your area, as they might help your family very much.
It is likely your SIL feels very badly for her illness and hurts deeply. She may feel very lonely, very tired of her illness, even suicidal. If you read this forum, you'll see how people with a bipolar diagnosis feel, how they struggle, etc. Reading here may give you a better understanding of your SIL.
I am sure your brother, married to your SIL, is also looking for some healthy support. He needs his family to support him and his wife.
I'm encouraging you to give this all some thought. I think you will benefit from setting some helpful boundaries so you are not drawn into altercations with your SIL. Clearly let her know that you care, yet cannot serve as her punching bag. Stop and think about the message you truly wish to give her, from your heart.
A "family intervention" would likely be helpful, as patterns of behaviors are already in place and are likely not as constructive as they could be -- for everyone. I am not sure a "family intervention" is a reasonable expectation as there aren't many services which help with interventions.
Again, think through the possibilities. What's truly helpful? How does everyone in the family feel accepted, heard and loved? Is it possible?
Again, welcome.

You are seeking some important answers. I hope we can help in some way.
Much love to you and to your whole family.
May love prevail.

WC