Is it permissible to take a percentage of a project's value in exchange for working on it, given the doubt surrounding the source of payment and the details of the contract between the concerned parties? And is the local official's knowledge of this sufficient, or must the funding entity also be informed?

If the project manager is authorized to employ whomever he wishes from the workers, then there is no blame upon him for agreeing with you two on the work, provided that the contract between you and him is legitimate. The safest approach is for the agreement to be on a known wage for a specified period or for a task to be completed. As for partnership or making the wage a percentage, that entails Shariah-related concerns.