I personally think this is a great question, and surprisingly one that has not yet been fully answered to my knowledge! There are two works that come very close to putting together a first-principles derivation that is fully quantum-mechanical, i.e. by quantizing the modes of a beam-splitter, showing their connection to a Hamiltonian and taking a limit that reproduces the unitary transformation you're referring to.
The papers are:
"Quasi mode theory of the beam splitter-a quantum scattering theory approach" by B. J. Dalton, Stephen M. Barnett & P. L. Knight
"A quantum scattering theory approach to quantum-optical measurements" B. J. Dalton, Stephen M. Barnett & P. L. Knight
In my advisor's group we're currently working on a tutorial that will go through this quantization and derivation to make it hopefully more clear - I'll try to post back here after we're done. A related question is how precisely to quantize a cavity coupled to the world - Sergio Dutra worked on this problem and distilled it very nicely in his book "Cavity Quantum Electrodynamics".