The nonuniqueness of Leray--Hopf solutions to the unforced incompressible 3D Navier--Stokes equations is one of the central open problems in mathematical fluid dynamics. Inspired by earlier works by Jia-Sverak, Guillod-Sverak, and Albritton-Brue-Colombo, we construct a Leray--Hopf solution in the self-similar setting and then establish the existence of a second solution by analyzing the stability of the linearized operator around this profile, showing that it corresponds to an unstable perturbation. To achieve this, we develop an innovative numerical method that computes candidate solutions with high precision and propose a framework for rigorously establishing exact solutions in a neighborhood of these candidates. A key step is to decompose the linearized operator into a coercive part plus a compact perturbation, inspired by the works of Chen-Hou, which is further approximated by a finite-rank operator up to a small error. The invertibility of the linearized operator restricted to the image of this finite-rank approximation is then rigorously verified using computer-assisted proofs.