How to generate graphs with a Hamiltonian path?
The simplest solution is the one suggested by Yuval Filmus. That is, take a simple path on $n$ vertices, and finish the construction after you have added some number of edges. Clearly, it is easy to enforce the maximum degree condition as well.

Alternatively, you could generate graphs whose structure guarantees the existence of a Hamiltonian path. For example, as shown by Tutte, every 4-connected planar graph has a Hamiltonian path (see e.g., Wikipedia for the statement). Then, a high-performance tool for generating planar graphs is plantri. My feeling is that this is unnecessarily complicated for your use case, but it gives you a possible another approach you might be interested in later on.