I find myself reading your post in a break from drafting a project plan. I cannot help feeling that I am just shooting in the dark, if not shooting the team in their feet.

The only tasks I can estimate with some level of accuracy are those where we are already half-way through.

I am working in an environment where the value of what the team is doing is pretty clear and we are just trying to bend all the project & budgeting rules to deliver that value.