Does there exist a type system for a non-let-polymorphic lambda calculus?
I am not sure what you mean by type system being decidable. I guess you mean type checking and type inference. (strictly speaking type inference is not a decision problem, although it can be casted as one and people usually think of it that way). 

In addition to intersection types, I would also suggest looking at System F where universal quantification does not appear only at the let construct. For system F, type inference is undecidable: paper