FEC and fluke need to be treated carefully. There are two issues.
If your vet does a normal flotation FEC she is unlikely to spot fluke eggs. The eggs collapse when the flotation solution is added. She would need to a special FEC, sedimentation, to check for fluke eggs. They sediment beautifully and if there are eggs in the faeces there's no reason a FEC wouldn't pick them up if done right.
The second issue is are the eggs actually in the faeces. Fluke don't shed eggs until they are mature, so for the first 9 or so weeks of infection you can quite happily get no eggs in faeces despite your animal being over-run with fluke. Once the fluke are adults they can shed eggs a little intermittently. Practically, this isn't an issue unless you have a very very low fluke burden. If you do have such a low burden it probably isn't doing your animals any harm (I'm talking 10 or less fluke in the liver).
There is a blood test that can be done to check for antibodies for liver fluke. However it only really tells you if the animal has ever been exposed. If you know the animal has never been treated for liver fluke then a positive antibody test will mean that the animal likely has fluke (although if the animal is old enough it could have had fluke that have then died). If you don't know the treatment history then a positive test doesn't tell you much. Likewise if the animal has only ever grazed on your land and it comes up positive it will tell you that at some point there was at least 1 fluke on your land, but if the grazing history is unknown it doesn't tell you much.
If you don't trust your vet with the FEC or any testing, you can send samples off to get tested. AHVLA and SAC do FEC for fluke. If they offer you a new 'coproantigen ELISA test' don't take it. It is costly and doesn't work all that well in horse samples.
In terms of is there a licensed treatment for horses. I can quite happily accept that there isn't one. It costs the drug companies money for each drug they licence for each different animal (they have to do all the safety testing). This is the reason that despite goats getting the same worms as sheep there are no treatments licensed in the UK for goats. If it's not commercially viable for the company to go through the testing to license it then they won't do it, vets will just prescribe off licence instead.
Hope that helps.
Dans