I believe you are overly ambitious in thinking you will find a projector in that budget range to get anywhere near that screen size with a satisfactory picture outdoors.
I suggest that you do some more reading on these forums and elsewhere to become more familiar with the challenges of what you are trying to do. I think by doing so you will be able to answer many of your own questions and you'll come up with new questions where people here can really help you.
As it is now, I think you are asking for too much from this setup and that makes it more difficult to help you. Sorry.