I would be reluctant to use a flip up cover – if anything got on it, like wet leaves or a cat deciding it was a great way to get off the wet ground, it could lead to strange intermittent problems although it probably would not lead to damage.
Normally an HEPA filter has a lot of folds or waves in it so it has more area and passes the same amount of air. If a repair guy was looking at the version you bought and making a correct judgement, it might be the case, but they sometimes say it on the basis of “I don’t know, so I will say the safest thing.” You could ask the manufacturer.
Again, it is unlikely to damage your AC, but if it restricts the air flow, then the chilled air available to cool your house might be too little to cope with the heat so the house would be warmer and the AC run longer, raising your electric bills.