Have you actually asked the landlord? If one of my tenants told me the bin I was providing wasn't big enough I would buy another (or reimburse them for doing so.) No harm asking.
If the landlord has already refused though, and the council don't provide them, then I'm afraid it's down to you. But we're not exactly talking a lot of money! Based on your hourly rate I bet you may have spent more time on this forum than the cost of a bin!
IANAL (I am not a lawyer).
Anything I say here is just an opinion, so should not be relied upon! Always check your facts with a professional who really knows their onions.