I think some clarification might be needed as there seems to be a bit of confusion between a "compost bin" as opposed to a "green garden waste bag" and even the brown (well, brown here in Bedfordshire) kitchen waste bin. For rabbit poo/bedding only the compost bin is really suitable - this is the cone shaped bin that stays in the garden and does not get emptied by the council. I think you might get away with putting straw in the green garden waste bag as it's in the "grass cuttings" category but not rabbit poo or you will evoke the Wrath of the Recycling Powers; you might be ostracized, excommunicated and exiled for non-compliance. And surely nobody would want to put rabbit bedding in the kitchen waste bin... I mean, it's not really food waste.
Also, personally I would not put any kitchen waste (like vegetable peelings) in the compost bin either; I did once and we ended up with a rat family moving in the bin - I lifted up the lid one day and there was mother rat nursing the youngsters. Unless your love for animals extends to having rats in your garden, I wouldn't do it. Since then, I have only put grass cuttings, soiled straw bedding and rabbit poo in the green compost bin and haven't had any problems. None of these are of any interest to rats.