I hated the sediment when I first started bottling Cider. I have now made a filter system that works great for me.
I have a filter that I got from toolstation for filtering heating oil (£4.50), I use this inline between two FVs, I stuff the filter full of cotton wool, I have also got a few tyre valves from a local tyre fitter, I have fitted these into the lid of a 25 litre plastic jerry can, I then fit the syphon pipe into the hole used for the air lock, I then use my foot pump to add pressure to the jerry can to push the Cider through the filter.
I know that oxygen can be bad for the beer/cider but it is only for a short time, I did try the little C02 cartridges but they are gone in a flash so I would use far too many of them, I cant afford a proper co2 cylinder at the moment.
I tend to syphon the Cider from the original FV after clearing a little with finings, I let it stand for a few days then filter twice, letting it stand a day in between. I find its really clear then but there is still enough yeast for bottle conditioning.
HTH.
Steve.