To be honest, I can't think of any type of bridesmaids dress that would be able to accommodate nursing without needing to be removed. Most dresses don't have "access panels" and are typically form-fitted, so the collar can't be simply pulled down.
Now is the time to go over your budget with your husband. Calculate the amount of $$ you will spend on gas for travel, the hotel & a few meals for you & the kids. Don't forget a wedding gift, and most likely a bridal shower gift as well! This is what you would spend, regardless of whether or not your family is "in" the wedding.
Then, calculate how much you can afford to pay for the "extras" of being in the wedding - A dress for you, tux rental for your husband, a dress for your daughter and either suit or tux rental for your son. Shoes to match, on all 4 (sometimes included in tux/suit rentals, sometimes not)
There are reasonable amounts to expect to pay for all of that. $200 for a dress may seem like a lot of money, but it is pretty average. You will need to find out if alterations @ a local shop would be included, if you ordered the dress from that shop, or else price out alterations with a dressmaker.
Looking online, tux rentals average between $150-200, with some higher/lower depending on the style & the store.
So right off the bat, you should figure $500 for the two of you to be in the wedding, plus additional expenses for your children. It is possible the bride would be willing for you to buy a dress in an appropriate color/style for your daughter, & you can probably get a suit @ a store for your son, vs. a rental fee. But let's figure another $100-150 for the two of them.
Keep in mind, these are reasonable, budget-minded expenses. When I stood up in my sisters wedding, she let us get our own dresses (from wherever we wanted) in a certain grouping of colors. On clearance @ Boston store, I found one, & still paid $110.
Additional expenses can include shoes to match your dress, jewelry to go with it, and salon expenses to have your hair/makeup done. All told, you & your husband should expect to pay $###-###-#### for the priviledge to take part in your brother & SIL's big day. For 4 people, that is not too expensive, although I do understand that a reasonable expense is always relative to the disposable income you have available.
Therefore, once you & your husband discuss whether you CAN afford to participate, you need to pick up the phone and call your SIL-to-be and have an honest conversation with her. If you can afford it, but it will be tight, let her know that. Say "I've talked with Jim, & we really want to be a part of your big day. Looking at the expenses for being able to attend, we have a tight budget of $XXX that we can spend on the outfits & accessories. I wanted to talk this over with you & see how we can make it work, so that you are able to have your dream wedding, and we are not stretched over our budget for the 4 of us).
Or, if necessary, say "I've talked with Jim, and after figuring the amount it will cost us simply to attend your wedding, there just isn't any way for the 4 of us to be able to participate in the wedding. By our tightest budget estimate, it will run $XXX, and we can't afford close to that. I wanted to let you know as soon as possible, so that you & my brother can make plans to adjust the wedding party accordingly."
In this case, you might even be able to offer "I know that the greatest part of that expense would be for Jim & I, but I thought maybe you & I could work out outfits that would not stretch us beyond our budget so that Sally & Johnny could still be your flowergirl & ring bearer."
I would personally leave the dress & breastfeeding hassle out of the conversation. Chances are, there isn't a dress that would easily accommodate your needs, & still be in fitting with the color & style of the rest of the bridesmaids. I'm assuming someone will be holding the baby for you while you are standing up during the ceremony, and when it comes time for feeding, you will need to make sure to have a quiet place to go. If possible, you could also pump & feed via a bottle (knowing that some people exclusively nurse & this may not be an option you are willing to consider)
Overall, you & your husband need to talk, & then you need to talk to your SIL-to-be, so that everyone is on the same page. Don't get hurt feelings over the nuances of your personal situation that do not seem to be taken into account - when planning a wedding, the different needs for a nursing mother are certainly not something a new bride is aware of, or understanding of how to accommodate. T.