Your sum is a fancy way of writing $$ A_j=A_j^{(4)}=\sum_{\substack{a_i\ge\lambda_i, i=1,\dots,4\\a_1+\dots+a_4=j}}a_1!a_2!a_3!a_4! $$ (which, of course, can be treated as $A_j^{(k)}$ for any $k\ge2$). If I understand you correctly, your expectation is that $A_j\sim N\times\{\text{the maximum term of the sum}\}$ as $j\to\infty$, where $N$ is the number of hits of this maximum term. This is certainly wrong, as the terms in the neighbourhoods of those maximal entries contribute quite substantially to the sum. Some plausible asymptotics to consider here are $A_{j+1}/A_j$ or $A_j^{1/j}$ as $j\to\infty$, as in these cases one can indeed show that the leading term completely determines the growth. The related reference to study is the book "Asymptotic methods in analysis" by de Bruijn (1961), more specifically, Chapter 3 there. (I would recommend doing $n=2$ and $n=3$ first.)

Your sum is a fancy way of writing $$ A_j=A_j^{(4)}=\sum_{\substack{a_i\ge\lambda_i, i=1,\dots,4\\a_1+\dots+a_4=j}}a_1!a_2!a_3!a_4! $$ (which, of course, can be treated as $A_j^{(k)}$ for any $k\ge2$). If I understand you correctly, your expectation is that $A_j\sim N\times\{\text{the maximum term of the sum}\}$ as $j\to\infty$, where $N$ is the number of hits of this maximum term. This is certainly wrong, as the terms in the neighbourhoods of those maximal entries contribute quite substantially to the sum. Some plausible asymptotics to consider here are $A_{j+1}/A_j$ or $A_j^{1/j}$ as $j\to\infty$, as in these cases one can indeed show that the leading term completely determines the growth. The related reference to study is the book "Asymptotic methods in analysis" by de Bruijn (1961), more specifically, Chapter 3 there. (I would recommend doing $k=2$ and $k=3$ first.) I really recommend this particular book, as the most accessible (and elementary enough) reference to the asymptotics of binomial sums.