Assume $n\geq t$ are positive integers and fix some arbitrary nonnegative weight vector $\vec\lambda = (\lambda_1,\dots,\lambda_t)$ where $\lambda_1+\cdots+\lambda_t = 1$. Suppose further that $\vec{n} = (n_1,\dots,n_t)$ is a weak composition of $n$ into $t$ parts which is nearly proportional to $\vec\lambda$ in the sense that $\|\vec{n}-n\vec\lambda\|_\infty<1$. In other words, $\vec n$ is obtained by rounding the vector $n\vec\lambda$ to an integer vector whose entries also sum to $n$.
Using Robbins' version of Stirling's approximation which states that $$ e^{ 1/(12n+1) }\leq \dfrac{n!}{\sqrt{2\pi\,n}\cdot (n/e)^n} \leq e^{1/(12n)}, $$ it follows that $$ \exp\left\{ \dfrac{1}{12n+1} - \sum_{i=1}^t\dfrac{1}{12n_i}\right\} \leq \dfrac{{n\choose \vec n}} {\sqrt{ \dfrac{n}{(2\pi)^{t-1}n_1\cdots n_t} } \cdot \dfrac{n^n}{n_1^{n_1}\cdots n_t^{n_t}} } \leq \exp\left\{ \dfrac{1}{12n} - \sum_{i=1}^t\dfrac{1}{12n_i+1}\right\}. $$ These upper and lower bounds only depend explicitly only on $n$ and $\vec{n}$. With some extra work, they can be replaced with bounds which depend only on $n$ and $\vec\lambda$, and with some additional work, they can be made to depend only on $n$ and $t$. However, we have not yet estimated $n_1^{n_1}\cdots n_t^{n_t}$. Clearly it should be close to $(\lambda_1n)^{\lambda_1 n}\cdots (\lambda_tn)^{\lambda_tn} = (\lambda_1^{\lambda_1}\cdots \lambda_t^{\lambda_t})^n\cdot n^n$, but my best efforts have not yielded an improvement on the following (rough) estimate. For convenience we write $(\varepsilon_1,\dots,\varepsilon_t) = \vec\varepsilon = \vec n - n\vec\lambda$ and abusively write $[a\,\dots\, b]$ to mean any number between $a$ and $b$: $$ \prod_{i=1}^t\dfrac{n_i^{n_i}}{(\lambda_in)^{\lambda_in}} = \prod_{i=1}^t \left(1+\dfrac{\varepsilon_i}{\lambda_in} \right)^{\lambda_in}\cdot n_i^{\varepsilon_i}\approx \prod_{i=1}^t e^{\varepsilon_i}\cdot (\lambda_in+\varepsilon_i)^{\varepsilon_i} = \prod_{i=1}^t[1/n\, \dots\, n]^{|\varepsilon_i|} = [n^{-t}\, \dots \, n^t]. $$ Is there a way to replace these $n^t$ and $n^{-t}$ estimates with numbers much closer to $1$? I do not mind if they depend on $n,\vec\lambda,t$. Note that I am seeking global bounds for all $n$ and $t$ and $\vec\lambda$ and have explicitly avoided big/little-oh notation (except for the abusive $\approx$ at the end, which I hope to eliminate as well).
Feel free to avoid Robbins' estimate as well. I just find that it does a nice job reducing to a cleaner problem.
EDIT: Technically it should be specified that we wish to explicitly estimate how close $n_1^{n_1+1/2}\cdots n_t^{n_t+1/2}$ is to $(\alpha_1n)^{\alpha_1n+1/2}\cdots (\alpha_tn)^{\alpha_tn+1/2}$.