You can try the following.

**EDIT:** At this point, I don't think this can work in full generality. However, it will give you some bounds on the singularities, which is better than nothing.

Take a resolution of singularities obtained by blowing-up a sequence of smooth centers $X_N \to X_{N-1} \to \dots \to X_1 \to X_0 = X$. Fix an $i$ you care about. Then by Corollary 1.4.3 of LINK: BCHM, you can contract all the exceptional divisors of $X_N \to X_i$ and obtain $X_i'$, something that agrees with $X_i$ in codim 1 (ie, up to flips/flops).

Then you could look at the largest coefficient you can stick on $E_i$ the (possibly non-prime) exceptional divisor of $X_i' \to X_{i-1}'$ (maybe that's only a rational map, I'm not sure) as you form this relative minimal-model. If you can stick a coefficient of $1-\varepsilon$ (for $\varepsilon > 0$) in this process then that should force $E_i$ to have nice singularities. In particular they should have semi-log canonical singularities and thus Du Bois singularities by LINK: KK (LC-singularities are Du Bois). The point is that a limit of klt singularities should be log canonical, and if you have a log canonical pair with (some) divisors of coefficient 1, then those divisors should be ``semi-log canonical'' for a sufficiently general notion of semi-log canonical. In particular, it certainly follows from
LINK: KK that such singularities are automatically Du Bois.

However, even if you can't manage the $1-\varepsilon$ bound, the bounds you can put on the coefficient might be useful to bound the singularities that appear (ie, you would effectively be bounding the log canonical threshold).

Again, this is very vague and I don't know if the details can be made to work.

Maybe Sandor will have some better thoughts.