A friend is very clued up on the law, and when confronted with an RPI who didn't want to show his ID, he immediately cautioned him. My friend read him the standard PACE caution, then informed him that he would be taking a statement from him about why he didn't show his ID when requested to do so.
The RPI went, in a nutshell, absolutely beserk. He threatened my friend with prison and all the rest of it, and it was all written down and noted. After the RPI came to his senses and realised that my friend was making meticulous notes, he backed down and offered to let it go. My friend, smelling blood this point, told him that it had gone too far for that, and that he wanted him to check and sign the notes so that further action could be taken. The RPI stormed off, but my friend then went and obtained a copy of the CCTV while informing the TOC that he was taking the matter further.
A week later, he got a phone call from a lawyer at the TOC, asking him to drop it and that they would offer adequate compensation for the distress caused. The adequate compensation turned out to be a free season ticket for a year, which tells you just how much he had them spooked.
A passenger is quite within their right to check the ID and verify it by any means necessary.