Before get this problem solved, which takes time, you and baby move to guest room to sleep. Depend on if your hu*****and is light sleeper or not, he can sleep in main bedroom with ear plug or sleep in other rooms, e.g., living room. Move the mattress there. (Or buy a really fancy air-mattress from Walmart for $50.) (You cannot use ear plug as you need to wake up for the baby.)
Unless your hu*****and works 10~12 hours per day, taking care of 3 months old baby is much harder than work in a company. You need good sleep more than he does. Your health and mental state are directly related to your baby's health and growth. If the baby is not doing well day one, you will have more challenges in the future.