Your residence card does not suffice for visa-free transit in the UK.
The Guide from the UK government to airlines mentions explicitly the need for a common format residence permit, that is, a credit card-sized plastic card or sticker in passport with either a bull (old format) or an European map (new format) on it.
The only residence permit issued pursuant ...
If I were you I would get a visa.
The document you have is not an EEA family permit, which was the name the UK gave to the documents that it issued like the document you have, but the UK never used that name to denote such documents issued by other countries.
The document you have is not a common-format residence permit. It is certainly a residence permit, ...
Yes, according to https://immi.homeaffairs.gov.au/visas/getting-a-visa/visa-listing/transit-771/travellers-eligible-to-transit-without-visa
Lebanese citizens are not eligible to transit without a visa.