We did baby signing. Never went to a single class, the book that initiated all this revolution was less than £10 and there are far more cheaper books on the topic today.
You are teaching sign language not BSL, she will learn to speak before she learn the first 20 sign and honestly, she will not need more than that.
By the way, the book approach is sooo good that the woman who was teaching babysigning classes locally (and charging a fortune for it) once told me that she was learning everything from the same book as I did, she just ensured that she was one week or two ahead of her class 
If your child has hearing problems, by all means pay a fortune otherwise the book will do.
Raising a child in a separated family setting is very very different to raising a child in a single home with two loving parents that find it easier to discuss and agree on expenses.
There are many things he should do to balance the amount of care you both provide for your kid but I know that deep down you know this guy cannot be expected to agree on how you want to raise your child.
You have years ahead of you, choose your battles, if you get in a pickle at this time about unnecessary expenses as decorating a room or baby tuition, by the time she is 3, you will be barely talking to each other (if he is still in the picture, according to Gingerbread almost half of non resident parents have little or no contact with their children two years after the split).
Choose your battles, and choose the important ones. The best thing you can do for your child is allowing her to grow in an environment where her parents are reasonable towards each other and can effectively co parent without having a war exploding regularly over menial stuff.