Oh Liv how sad! Poor little one. I know you were worried about money but even if you had all the money in the world it sounds like you are doing the most humane thing anyway.
Big hugs to you and your family. I would include your DD in the burial. Our DS is the same age and our family dog died last year. He is buried in our yard. DS wasn't there because we were living away at the time but he knows that Bundy died and where the grave is. He talks about him and adjusted quite well. I think it is easier for him to understand than if he was just gone.
Thinking of you. xo