No. You just THINK there is some obscure MATLAB function to solve your problem. You do not know that to be the case, because there is not. You just want there to be an existing code to solve your problem.
Can you write some version of Levenberg-Marquardt? Of course not, since it would require computing a derivative.
My suggestion would be to use a guarded method, where you start with two points that bracket the solution. (Much like fzero.)
Now try to use a secant method, where simple linear interpolation is employed between those points, rounding the new identified point at each step. If the new point chosen is too close to either endpoint, then you want to resort to ideas like modified regula falsi, or even bisection. Again, at each step, you will need to round the result. But the idea of a guarded scheme is important, since it will be able to converge well in regions where the function allows it, but it has the ability to handle the bad cases robustly.
Essentially, you almost want a version of fzero that worries about an integer independent variable. That is because fzero is the code you would want to use, except fzero is not designed to work with integers.
