"My manager agreed to let me get promoted."
I think it is a mistake to translate 让 as 'let' here. l think the sentence would be better translated by either 1. My manager agreed to have me promoted; or perhaps 2. My manager agreed to get me promoted.
I don't think you need to say 我的 at the beginning... It seems a tad redunant.
I have learned that 的 may only be omitted for close familial relations.