Thanks for pointing it out. I've updated the post.

I think you missed out something important: $$$C / gcd(A,B,C) \geq 200 \Rightarrow C \geq 200$$$. Since $$$Cz \leq D$$$, that means that $$$z \leq \frac{D}{C} \leq \frac{D}{200}$$$. Therefore, you can try all possible values of $$$z$$$ (from $$$0$$$ to $$$\frac{D}{200}$$$), and for each value of $$$z$$$, you can calculate the number of solutions of $$$Ax+By = D-Cz$$$.

I'm still interested in the solution for the general case of the problem. Using generating functions, I know that the number of solutions should be the coefficient of $$$x^D$$$ in $$$(\sum_{i=0}^\infty x^{Ai})(\sum_{i=0}^\infty x^{Bi})(\sum_{i=0}^\infty x^{Ci}) = \frac{1}{(1-x^{Ai})(1-x^{Bi})(1-x^{Ci})}$$$, but I'm unable to find a closed-form formula for that coefficient.