To what extent is it permissible for an engineer working in a consulting office to design buildings belonging to a contractor who is not a client of the office, given that the engineer is responsible for supervising the projects of this contractor through the consulting office?

You are not permitted to undertake the design, because one of the contract's stipulations is that the contractor commissions the design from another engineering office. Also, Muslims are bound by their conditions, and suspicion would arise from your follow-up and supervision of your own work.