MHLib.jl may indeed be a good starting point for your work when you want to apply a heuristic approach. To realize a basic (general) variable neighborhood search, GRASP, (adaptive) large neighborhood search etc. is easy for a black-box permutation problem by just updating the TSP demo, see the more detailed comments at here.
Note, however, that as for any challenging combinatorial optimization problem, it always is a good idea to think about the problem structure and to possibly realize more specialized methods you can provide to the framework. For example, a problem-specific construction heuristic for creating an initial solution might be a good idea, or some more specific neighborhood structures, possibly allowing an incremental solution evaluation. MHLib.jl only contains basic functionality, but is designed for efficiency and to give much flexibility for such problem-specific extensions.
Disclaimer: I am one of the authors of MHLib.jl.