You should not expect that a professor that you don't know at all will give you a project to work on before you have accepted an offer at their university. I would say the comment to contact him when you apply is so that he can see what your application is like before committing to anything - I wouldn't read much into it.
I'm a bit confused by your post. Are you expecting to get a LOR from a professor you don't know who you just email out of the blue? I think you would be extremely unlikely to find anyone who would say yes to such a request, and even then, how is it going to be better than the letter from your department? "Barely knows me" is better than "doesn't know me at all, we haven't even met". Also, don't focus on getting a letter from a famous person - if they don't know you at all, that letter will be worthless. It's better to have a letter from somebody who actually knows you, at least a bit.
If you are not applying for a few months, you could try to get to know your 3rd recommender a bit better in the mean time. I would say you should still apply to other universities with your current 3 letters, though. You may get accepted anyway!
Why would you stay at your current university? (E.g. would you be getting a Masters degree?) Do you mean extending your bachelors one more year? Unless you are actually interested in this professor's area, it seems like it's not a good use of your time.