I think it is a good idea to do your best to persuade school board. You can consult a lawyer to see what option you have. However, I don't think it is a good idea to sue the school district as I believe it is much cheaper to send your daughter to a private school.
When I was trying to make decision whether to send my summer birthday boy, who just met the cutoff date, to K, I had him evaluated by an early education expert. She is not a psychologist. The process is quite painless to my son. (Some organization provides free service for this. I am not in CT.) Another question, if you think it is not good for your daughter to be evaluated by a specialist as she knows what is happening, won't be an even worse situation for her to repeat K while her friend goes to 1st grade? BTW, you can talk to your PD to see if she has any recommendations.
Good luck.