Local Search Optimization Algorithms in Applied Settings -- How best to explain to applied researchers?
The question is a bit unclear, but just to add my two cents:

Simulated Annealing and Genetic Algorithms are not classified as ”local” optimization algorithms. They have been designed to overcome the limitations of local searches so they are usually put in the “global” optimization basket.
The fact that they are stochastic doesn’t mean that their results are not reproducible. Standard implementations of SA and GA allow you to specify a random seed, so the optimizations are perfectly reproducible.

So you have many local optima from your algorithm. Do the “better” local optima tell you anything more than the worse ones? What is the actual, physical meaning of your optima? Is there any way you could compare your algorithm with other, established optimization procedures to assess the quality of your implementation? How many parameters (optimization variables) does your model have? Is your objective function expensive to evaluate or quick? We haven’t seen your model so it’s difficult to suggest anything about possible arguments to convince your colleagues.
That said, having an objective function full of local optima is nothing new, very many real life applications are like that. However, in almost all cases we strive for better and better optima all the time (I.e. lower objective functions in minimizations, higher for maximizations) and when we have to account for many optima (for example, for forecasting a matched model) we usually have a good, physical explanation for what we are doing.
I have to admit that your case is a bit unusual, as having orders of magnitude different optima is unheard of (at least for me). But since you haven’t provided a model nor benchmarked your algorithm against others I am afraid these is the best suggestions I can come up with.