I agree with Miss Lydia. You should always put landscape fabric under gravel or the gravel will just be pushed or washed deeper and deeper into the soil until it is all gone. I don't have any duck experience though but I don't think it would be a problem with webbed feet. As far as draining, with 40 odd years in the landscape industry I haven't know the fabric to impede the drainage.
Water drainage isn't a problem....mine is very heavy duty but it still lets the water through (though much slower than normal) but it doesn't allow the poop to wash through so you get a 'mat' of poop between the gravel and the fabric and it can get quite unpleasant! This was with quail....and ducks poop bigger and more copiously and also have a lot of water around so you could end up with a yucky 'poop soup' in there

OP....I would definitely go with hardware cloth, just make sure it's small enough mesh to not allow the gravel through xx

. They definitely did stuff like that here. You get excavating, always turning up something. Adventures! 