Since the map projection is dependent on complex parameters not exposed to us, a formula-based exact solution seems unlikely.
Instead, I'd recommend the game developer's favourite tool: piecewise approximation! This is how we handle textures, 3D models, time and physics, and it works for maps too.
To simplify the problem, we'll assume
y = elevation (you can add a scaling or offset factor here as needed), and focus on how (lat, long) map to (x, z) in your scene.
Our strategy will be to cover our map in a web of known points. Then for any arbitrary input within the span of that web, we can interpolate the position of nearby known points.
Start by manually picking feature points that you can clearly identify in both your mapping application and your Unity scene built from that map. Make sure you have good coverage in your main areas of interest, and have at least some points all the way out to the outer bounds of your Unity scene / outside the expected area of play.
For each point, note both:
Use Delaunay triangulation to turn this point cloud in the 2D lat-long plane into a triangle mesh.
Now it's effectively an inverse texture mapping problem: we have a set of vertices (x, z) with texture coordinates (lat, long), forming a triangle mesh over your Unity scene. Given a texture coordinate (lat, long), we want to know where it sits on one of your mesh's triangles.
We can use a Binary Space Partition to divide the lat-long plane into polygons along the triangle edges, each touching three of your feature points. You can then walk the tree to find which polygon a particular input (lat, long) pair falls into, in logarithmic time, and get the three feature points for that region.
Next you can find the barycentric coordinates of this (lat, long) pair compared against its feature point triangle's (lat, long) positions. Use these barycentric coordinates as interpolation weights to blend the feature points' (x, z) positions, then substitute the y you got from your elevation, and the result is a 3D position in your Unity scene.
The position you get out will exactly match your map at the feature points, and use an affine approximation in between. You can add more closely spaced feature points if you find it's inaccurate in any problem regions.