Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
The closest jurisprudential characterization for the operations of the aforementioned construction company is a Liabilities-Based Leasing Contract (Ijarah al-Zimmah). In this arrangement, the company leases the services of the workers for a specific duration at an agreed-upon wage, and subsequently sub-leases these services to another party at a higher rate. The company assumes the workers' expenses, including visa fees, housing, food, drink, and transportation.
Imam al-Kasani al-Hanafi (may Allah have mercy on him) stated: "An employee (contractor) may carry out the work himself or through his own employees, provided the contract does not stipulate that he must perform the work personally. This is because the contract is based on the completion of the work itself. A person may fulfill an obligation either personally or through others, as the work performed by his employees is attributed to him, making it as though he did it himself. This applies unless personal performance was a condition of the contract, as the agreement would then be tied to the work of a specific individual." [Bada’i al-Sana’i, Vol.4/P.208].
As for the condition set by one of the managers of the external company to receive a percentage for providing the workers, this is impermissible. Since he is an employee of that company, he is not allowed to take an independent percentage or commission beyond the salary agreed upon with his employer, except with their explicit permission.
In conclusion, there is no Shar’i objection to the business model of the construction company as described. However, it is not permissible to grant the aforementioned manager any commission outside of his official salary from his company unless they authorize it. And Allah the Almighty knows best.