Is it permissible to deal with public sector employees who demand a share in profits in exchange for facilitating the completion of projects, knowing that this dealing might be on the verge of speculation (gharar)?

Mudarabah (profit-sharing partnership) is valid if it fulfills its conditions and its impediments are absent. Among the conditions for the validity of a partnership is that there should be no compulsion on either partner. If there was compulsion, the Mudarabah is not valid unless the coerced partner willingly consents. If the transaction is not permissible by Sharia, your friend must seek what is lawful and look for another job. A man with such experience will not fail to find another means of livelihood, as Allah Almighty says: ﴿And whoever fears Allah – He will make for him a way out And will provide for him from where he does not expect.﴾ The rule "necessities permit the forbidden" does not apply to your friend's situation, as he can look for another job. For "necessities are estimated according to their extent" and "what came for an excuse becomes void when the excuse ceases." Moreover, the jurisprudential rules you mentioned, such as "earning with some doubt is better than begging people," do not apply to your friend's situation, as he is not compelled to beg people. Your friend should bear in mind the rule: "Seeking lawful earning is an obligation after the obligatory duties." It is not permissible for a capable Muslim not to engage in a lawful and honorable work from which he earns his living and sustenance.