I think your proposal might be the best that conveys the idea of convincing someone to be nice to you without being manipulative. In my personal experience (native French, 25y/o, educated middle-class background), this expression can be understood by almost anyone and does not sound formal as it is an expression.

Something that might sound a bit too shady is être dans les petits papiers [de qqun]. This sounds a bit like the boss would give unfair favors to the employee because of that.

Il essaie d’être dans les petits papiers de son patron en proposant de faire des heures supplémentaires.

Another expression that came to my mind is mettre [qqun] dans sa poche.

Il essaie de mettre son patron dans sa poche en proposant de faire des heures supplémentaires.

or

Il essaie de se mettre son patron dans la poche en proposant de faire des heures supplémentaires.

However, in this context, I’m not sure it’s appropriate as it involves a form of manipulation, or maybe even superiority, over the person you are trying to convince.

You can go for something like :

Il cherche à attirer les faveurs de son patron en proposant de faire des heures supplémentaires.