PHP Script for RSA Key, JWT Signing

I need a PHP script capable of generating and signing JSON Web Tokens (JWT) offline. The signing process should replicate the behavior of an existing system where the private key is inaccessible. The goal is to simulate the signing mechanism without relying on direct access to the private key.

Requirements:

Develop a PHP script to interact with a given API endpoint that requires signed JWTs for authentication.
Analyze the target system's logic to understand its signing requirements.
Implement a solution to generate valid JWTs based on the system's structure and expected behavior, bypassing any dependency on direct private key access.
The script should include:
JWT header generation (e.g., alg: RS256, typ: JWT, kid: key ID).
Payload creation based on specific input requirements.
Signature generation to match the target API's expectations.
Key Deliverables:

A fully functional PHP script that produces valid, signed JWTs accepted by the target API.
A concise explanation of the approach and methods used to simulate the signing mechanism.
Additional Details:

This project is for research purposes and requires a strong understanding of cryptography, PHP, and JWT workflows.