YANBU...I haven't read the whole thread (I usually do, but I'm a bit short of time).
My dd is now 8 and a very fussy eater. I try really hard to get her to eat other stuff and I know how it is when they won't eat something on the menu of a standard restaurant. Mine will eat pasta but not pizza and she'll eat chips, so she's ok in most places, but I still feel for the fact that she'll struggle with invites like yours in years to come and am constantly pushing her to try new things in an attempt to avoid this.
If it was my child that had been invited and I knew how fussy she was I'd either say no, which is what my dd would probably choose, or I'd say you can go, I'll feed you beforehand and you can just have some pud at the end. I'd tell her to be on best behaviour during the meal and I'd explain this to the parents. I'd also be very, very apologetic for my daughter's lack of flexibility.
I feel for you both and some of the responses on here just reaffirm to me the way my own dd is viewed. It's a tough issue, much tougher than some parents think.
We've bullied, cajoled, encouraged, urged, bribed and downright deprived dd of other foods. Nothing works. Now, we relax and just make sure that something new goes on the plate every night. Most days I make her take a taste at least a little. Some days I don't push her.
Some parents will always judge the parents of a child like this. Please don't join in the mom bashing. Just find a way around it that doesn't affect your child's day or yours. There will be a solution. Have a great day.