Is it permissible for an employee of an organization that grants funds to projects to accept dinner invitations from beneficiaries, especially if some of them might offer him personal services or intercessions?

You have done well in refusing these invitations, as they fall under the category of prohibited gifts to employees. The only exception is what is offered purely as hospitality. There is no objection to utilizing the services of these individuals, provided that it does not lead to favoritism towards them at the expense of the work, and with the necessity of adhering to the Shariah-compliant regulations for mediation.