Even with BEM you still need to learn how to generate a (surface) mesh, although in 2d this is just lines. Really, there’s not much alternative to learning Gmsh or some similar package in the long run.
I agree that BEM or related surface integral equation methods (e.g. Nystrom) could be the most computationally efficient methods for this problem, but they are finicky to implement well and have a steep learning curve. I suspect it will be a lot quicker (in programmer time) just to bite the bullet and learn Gmsh and Gridap, unless BEAST already solves exactly the problem you want. The problem you are describing is not too computationally demanding.