No way is needing to clean the house bad enough for bullet biting, as should be obvious from the derivation of the phrase.
"To bite the bullet" is quite literal; it refers to very badly injured soldiers, far from any proper help and needing immediate, radical treatment - such as having a leg amputated without anaesthetics or antibiotics.
Since there is nothing else to hand, the soldier - or explorer or whatever kind of particularly rugged life-form we're talking about - bites a bullet because often, biting very hard on anything you care to mention will distract the patient from the pain being inflicted on him. Biting on anything also makes it difficult to scream and so distract the rescuers…
Anything needing someone to "bite the bullet" will be extremely painful or extremely dangerous or both. Cleaning the house is merely unpleasant.