So essentially 5mins after take off, the single planned alternate, LCA, was no longer practicable...
Essentally correct, not because of the expense but probably because LCA lacked an (AF) approved maintenance organisation, meaning the aircraft would have been stuck there until the parts and a certified mechanic could be flown in.
However I think I would have made the same decision, if the MEL says the flight may be continued to the next landing and the destination has suitable maintenance facilities. At the time everyone involved could safely assume that the flight will continue to Beirut.
The fundamental mistake was the decision to divert to AMM without making sure that all the necessary paperwork was in order. Anyone operating in that part of the world should know better than to simply redespatch in mid-air (...at least they did not consider TLV, they did have some vague ideas after all...). Clearly it would have been a better choice to bite the bulet and go for LCA and either organise the maintenance support, or get a one-off waiver for the next flight from the DGAC (it seems they eventually did for the LCA-BEY leg - I don't think the issue was ever raised leaving DAM).