Appealing against a competency matrix

Hi everyone. I have been at my place of work for 14 months. Everything was fine until I got a new manager (external candidate, the role was not offered internally). I was seen as the go-to guy in the department and made myself available and offered help and suggestions from his first day. He took alot of my ideas on board, but we never really built a rapport. He hired a contractor and I trained her and showed her the ropes. He then proceeded to give her additional responsibility only after 2 weeks. She says she doesnt know him from before but i found out that they came from the same agency. Now due to a restructure in my department, I have been put at risk of redundancy. There is only one position in the London office and two elsewhere. The role is slightly different and carries more responsibility.The contractor and I are the only two people going for the London role. My gut instinct told me that he was gong to give her the job.We were not interviewed but they carried out a skills competency matrix. I was told today that she scored higher than me. They have not showed me my figure or how i scored, just the competencies. They said that in some parts i scored a 2- falls below expectation but they didnt tell me where or how.She scored hire in the responsibility sections, because h gave her those responsibilities. I have the right to appeal and am trying to decide if i should. I get a months wage redundancy due to length of service. I don't want the job as I can't work with the manager, but I can't bear the injustice of it all. Also I recorded the conversation but didnt tell them. This was purely so I didn't miss anything as I was stressed and not taking it all in. Part of me wants to appeal to prove I deserve it more than her, and to highlight that they are just going through the motions. Can anyone offer any help on appealing against a skills matrix?