Human resources advice

Job ID: 36877837

Budget: £20 – £250 GBP

I have been working for a large US international company in the UK for 7 years now as a full-time vendor (agency) in the technical department. During my time here, I have successfully created and completed numerous global projects as extra outside my main job duty aimed at improving workflow and establishing new procedures. As a result, I have gained a reputation as a competent and respected employee.
Recently, I applied for an internal position within the company but was not selected. While I understand and accept the reasoning behind not getting the job, I am aware that some individuals with no experience in the field have been hired in the past. However, the issue I am facing lies with my manager, who has displayed toxic and manipulative behavior throughout the years. His conduct has been highly unprofessional and rude.
I have made the decision to address my concerns about my future within the company and how my manager's behavior is impacting it by writing an email to higher-level managers. Over time, I have noticed that my manager does not support my professional growth and prefers to keep me in my current position. He has been spreading negative remarks about me to people in other departments, yet when I inquire about my performance, he tells me everything is going well. Additionally, he has repeatedly promised me an in-house job while consistently bullying me. In the hopes of improving my chances for advancement, I have remained silent and avoided causing any disruptions.
It is important to note that I am not the only one who has experienced issues with my manager. The entire "in-house" team, including myself, has a negative opinion of him, but everyone is too afraid to speak up. If approached, my colleagues would be willing to confirm their thoughts, but they are hesitant to initiate such a conversation. Recently, my manager's supervisor has become aware of the negative feedback and has begun speaking individually with team members to discuss his unprofessional behavior. Last week, he even approached me and inquired about how I felt regarding not getting the job, providing a motivational speech. Towards the end, he hinted that if I had anything to discuss, I should approach him for a chat. It was evident that he wanted to discuss my manager.
Due to these circumstances, I have decided to write a letter to the supervisor, outlining the issues I have experienced with my manager. I am aware that my team members are reluctant to voice their concerns openly, but they have expressed willingness to confirm the information if asked. I would greatly appreciate the guidance of an HR professional to review my letter and advise me on any necessary modifications or additions. Furthermore, I would appreciate advice on how to handle the situation once my manager supervisor reads the letter and approaches me for a discussion. Some details may not be clear at the moment, but they will become apparent once you have read my letter.