If baby is bf then it is not approproate for them to try and give your ds a bottle. Firstly it will be alien to him and he may or may not take to it. If he takes to it he may refuse breast afterwards even, or it may simply be very traumatic for him. And then of course it also will interupt your milk supply. People who don't know think it's easy just to express some milk and off they go. It isn't necessarily so. He may think you are being difficult by refusing to express so you need to explain that it isn't that simple and some people simply can't express. And frankly why should you express and why should your baby be forced to take a bottle!
Does your ex realise what a bf baby means? No of course he doesn't. You need to explain to him that it means feeding on demand and it means not being away from mum for more than an hour in reality. It doesn't mean giving a bottle instead.
He really needs to go on a parenting course to help him understand. And you should suggest he gets the 'birth to five matters' book and read it.
And no twice a month is not enough for your baby to know who his father is. He won't recongnise his voice or smell or other factors which help reasure a small baby. So your ex is in effect a stranger.
You wouldn't leave your baby with a stranger unless it was someone you knew was trustworthy and was aware of the needs of a baby. So you may consider a childminder/nanny here, but all would insist on a taster session to see how baby got on and follow your instructions to the letter. Do you think your ex even knows what to do? You need to see him changing nappies, talking to and getting to know your ds before he takes him for a short walk (half hour). If he insists he wants to do this away from your house then tell him he needs to find a contact centre where there will be staff around to help him and reasure you that baby is being taken care of properly.
If he gets cross about this then you simply point out that getting cross at you for a perfectly reasonable request does not give you any kind of confidence he can be responsible enough to care for a small baby. Remind him he has no previous experience and you had your midwife, health visitor and family to call on to help you learn. Would he really expect you to hand over your baby to someone you are not entirely sure is capable of taking care of your baby and who isn't willing to prove that he is but taking the necessary time to get to know him and learn about being a parent.
I would suggest that your ex is obviously more concerned on getting out of the house and spending time with his new partner in conjunction with ds. I can see how this grates on you (why should she get to play happy families with him and your ds), but on some level, when the time comes, it might be better to have someone else there (depending on how responsible she is of course)
Again you need to point out that your baby's welfare is more important that his new partner's feelings and if she's not prepared to wait or be understanding of the situation then it doesn't inspie you with confidence that she or he are thinking of ds first and foremost.
Please don't be bullied into it.