ok, i don't mean to criticise. i am aware that what i say will not be easy to accept and isn't necessarily what you want to hear but this is based on my own experience.
at the moment i think you are absoloutely doing the right thing WRT introducing his father slowly and supervised until your ds is happy to be alone with him. i think letting the grandmother visit for an hour the evening before his birthday or the morning of, would do no harm whatsoever to your son. babies see new faces all the time and seeing one new one for a short time, with you there would be very unlikely to traumatise him. if he became unsettled you would be totally right to bring the visit to an end. any decent grandmother would agree with you that it was the right thing to do rather than upset the baby.
in the long run, things will get easier, your EX will show his colours as to whether he is going to stick around or not. hopefully he will realise his son is worth it and make the effort.
when i had my first son i was separated from my EX and after the first few weeks the novelty wore off for him and he stopped coming to see our son. all contact stopped and my son was 4 months old before he decided he wanted to be a dad and understandably i was resistant to this. i had been raising my son myself, with no help from him either physically or financially. i thought " who the hell do you think you are marching in here demanding your rights when i have been doing the donkey work and you couldn't have cared less if we were living in a cardboard box." i was angry. i didn't want to have to hand my son over to this person who was quite happy to disappear until recently. i saw a solicitor and she confirmed that he would get contact if it went to court so far better to agree something between ourselves. contact was agreed, starting off with supervised in my house progressing on to my EX taking ds to his own house. i remained very nervous. would EXP know how to care for him alone, would he drive carefully, would he fuck off when things got a bit uncomfortable and leave ds with his grandmother (EX's mum). i didn't liek any of it. i didn't like not knowing what would be happening to my son whilst i wasn't there. i didn't like not being able to be there is ds was upset or tired (and he was bound to have been both of those at various points during the contact). it was very hard to accept all this and i felt very justified in feeling this way, because afterall, i was his mother and i knew what was best for him didn't i? time went on and my EX stopped his contact and moved away. he saw ds sporadically over a 3 year period. ds knew who 'daddy' was but it was the sort of relationship he would have with a friend of mine, it wasn't a father son relationship. after 3 years EXP got in touch with me and we did alot of talking. he still lives away but between the two of us we have made teh effort to make sure ds has a relationship with him. i am now at a point where i respect his right as an equal parent to be as involved as possible with ds' upbringing. day to day stuff falls to me, we both agree on the usual stuff like food and discipline so it rarely is an issue but if something comes up that i think he should have a say in, i text or ring him and we talk. personally, i am a far happier person now that we have this type of parenting relationship. i resented being the one doing all the work, and whilst i am still doing most of it now, it somehow helps to know that the one other person in the world that loves ds like I do is there to share the big decisions with. as a parent we always want to do the right thing, as a single parent i often find myself thinking "oh god, is this really right? i would love someone to bounce ideas of" i have that now. we dont always agree but we try and talk things through and keep what's important at the front of our minds. i hope for you and DS that you can get to a point with your EX where you have that support. i hope your EX steps up to the plate and does right by your son. but most of all i hope that you and yoru ds are happy. this may be without your ex in his life and that may be what is best for your son. it will take time for things to settle down, your Ex will have to extablish a routine with you and your son and he will have to be very flexible while your son is so young. it will also require compromise on your part to allow their relationship to develop. good luck with it all. i know it isn't easy, but hopefully it will get easier.