The following might lead to answer to your question (I am posting it as an answer instead of a comment because it takes space). Someone once told me Peter May showed that a fibration is always fiber homotopy equivalent to some fiber bundle (where the fiber can be pretty large). I don't know how this is proved, and I am not sure if I am even remembering the statement correctly. But here goes...
Assuming it is correct, then to every fibration $E \to B$ with fiber $X$ which is classified by a map $B \to BG(X)$, where $G(X)$ is the monoid of homotopy automorphisms, there is supposed to be a homotopy equivalence $Y \to X$ and a factorization up to homotopy of the classifying map: $$ B \to B\text{aut}(Y) \to BG(Y) \simeq BG(X) . $$ So if we consider the universal (quasi-)fibration $$ F \to EG(X) \times_{G(X)} F \to BG(X) $$ with classifying map the identity $BG(X) \to BG(X)$, it will factorize through $B\text{aut}(Y)$ for some choice of $Y$. This would show at least that $BG(X)$ is a retract of $B\text{aut}(Y)$.
Note added: the result of Peter May I quoted is also contained in a paper by Casson and Gottlieb. They give a geometric argument along the following lines. Suppose that $E\to B$ is a fibration with fiber $X$ a finite CW complex. Then $X$ can be thickened to a compact codimension zero open submanifold $M \subset \Bbb R^n$ for some $n$ depending on $\dim X$. The derivative map $$ \text{diff}(M)\to F(M,O_n) $$ is then defined. Let $\text{diff'}(M)$ be its homotopy fiber. These are the diffeomorphisms of $f$ $M$ equipped with a path from $df\: M \to O_n$ to the identity.
Using Smale-Hirsch theory, it's possible to show that the map $\text{diff'}(M)\to G(M)$ is highly connected: the range of dimensions depends on $n$ and $\dim X$, but it tends to infinity with $n$ (where we stabilize $M$ by $M \times \Bbb R$ to increase $n$).
Consequently, if $n$ is taken very large and $B$ is finite dimensional, we can find a homotopy factorization $$ B \to B\text{diff'}(M \times \Bbb R^n) \to B\text{diff}(M\times \Bbb R^n) \to BG(M) $$ This shows that any fibration with a finite CW fiber over a finite dimensional base space lifts to a fiber bundle.