I suggest that you get an attorney. Trying to deal with the father on your own just feels too stressful to me. Go to court and get everything into a court order. By filing in court, you will have access to a facilitator who will help the two of you work out a plan.
As your baby's father, he is able to put him on his insurance plan. Perhaps he would have to pay extra. I doubt that you can put him on medicaid since the state knows that he's the father and has easy access to insurance.
I suggest that he doesn't want to go to court because he knows that in the end he will have to pay more. I urge you to go to court to get what your baby and you are entitled to having. I also suggest that this is insight into what life would be like if you were to marry him. Tight fisted people rarely change to a more open way of thinking. Being tight fisted is usually related to more than money.
Yes, it is important for your baby to have a relationship with his father and I agree that you are right in working towards that. I also agree that overnights are not appropriate at this age, especially since you are breast feeding. He does need the security of being in one place and being breast fed on schedule or demand.
I suggest that it could be possible that the father could be alone with the baby for short periods of time. How to arrange that logistically is complicated. Perhaps a compromise would be for you to let him have the baby in one room while you're in another part of the house.
As to court, the best interests of the child is their primary concern. You can ask for your baby to have his own attorney to further ensure that happen tho that may make the case more complicated. Given the two options, of trying to deal with a "tight fisted" father alone and having the court decide, I'd definitely choose going to court.
I have had much experience with the court system and do trust the one in my county. Perhaps there are or will be posters from your county who can reassure you. I suggest that you learn about your county's court system. You can usually consult with an attorney without charge as you work towards hiring one. A part of that visit is to determine if this attorney is one that agrees with your parenting style and would work towards your goals. You can also learn about what to expect in court as well as the cost and how you can arrange payments.
If you know that you cannot afford a private attorney, make an appointment with Legal Aid to see how they can help you. You don't have to use them. I'm suggesting that you explore all options.
I'm also suggesting that it is in both yours and your baby's best interests to have an unrelated third person with legal experience represent you. You can negotiate with this help and perhaps avoid court. A mediator could help you work with the baby's father to come to an out of court decision. You will benefit from getting professional help and that can be arranged without horrendous cost.
I wish you well at this difficult time. I am concerned that the father doesn't seem aware of the importance of consistent breast feeding. Perhaps you could provide him with written info about breast feeding as well as the needs of a 3 month old infant. There is much info on the Internet that you could send him via e-mail.
Later:
I met an ER nurse who flew to the East coast twice/month to spend time with his toddler son. Your baby's father could get a room and stay the weekend, thus having more time with his son. I understand you needing space to figure out what you want to do with this relationship and having the father in town for a weekend doesn't provide much space but perhaps it would be best for now. Perhaps a compromise would be to allow 1-2 bottle feedings/24 hours so that the father could spend a few hours with his son. He would also be in town which might help you feel more secure about your baby's welfare.
I urge you to get involved, either thru the courts or by hiring a private one, raw with a mediator. It sounds like both of you do want what is best for your son. A disinterested but professional third part could hasten this process and provide a sense of control for both of you.
Portland has an attorney's office that advertises mediation. It may not be easy to find a mediation service but I urge you to look. Talk with an attorney, too. This process is very complicated and could end up with no one being happy. Get professional help!
I strongly agree that whatever you decide, whether thru mediation or not, be put into a court order. The situation is much more clear to everyone involved that way.