Bedding

I am in a quandry

I have hay in the litter tray only as mainly Twinkle will poop/wee wherever there is hay. She also poops on vetbed... so what can I use for decent bedding that she wont poop/wee on when it starts getting colder?

At the moment there is just newspaper down and vetbed when it is cleaned and dry... but am struggling to find a bedding she wont poop on and worry that she will be cold in the winter.

I've had this dilemma with a few buns before. Is there any way you can just completely fill the sleeping compartment with hay? It'll be warm but only need cleaning every day or so, maybe even less. Otherwise you could just fill a cardboard box full of hay & tip it into the bin every night or somethign? X

What about shredded paper? That's meant to retain heat quite well I think?