When you buy a J ticket between the Gulf states and a long haul destination, you are buying a business class ticket and being accommodated in the First cabin on the short haul sectors as a courtesy because there is no business cabin. Effectively it's an op-up. You have a First Class boarding pass but have not bought a First Class ticket for those sectors, and so are subject to the restrictions of your underlying business class fare bucket - in this case P (Lite). This is also the fare bucket on which qpoints and avios will be awarded. QR appear to have applied the eligibility rules correctly here, however frustrating that may be.

A higher J fare bucket, or a full F ticket for the whole journey on routes where that is possible (not available to SIN unfortunately), would have resolved the lounge access issue. Unfortunately many third party agents simply list the lowest possible fare to get their entries higher up in search engine results and aren't always explicit about what may or may not be included. Your beef should really be with the agent for not providing fuller information.