I am of 50% Indian and 50% British heritage, and 100% British culturally. From that perspective, I will speak plainly.
The father in this story is actually fully Italian. He was born Italian, and does not speak the Senegalese language, cannot cook the cuisine and is unfamiliar with the music.
However, your son is the beneficiary of two rich, deep and creative cultures, British and Italian, instilled in him in the only way culture is or can be transmitted, purely by social interaction with his parents and environment, and that's plenty good enough.
Your child does not have African "roots". Rather, the portion of his DNA that codes for skin colour, promotes slightly more expression of melatonin. That's all, that's it. There is no magical "cultural" feature embedded in his DNA.
You do not inherit culture through your skin colour.
This is like telling me that because I have a slightly olive-coloured skin, I am somehow connected to India in some mystical way that other people aren't. Let's be clear: to claim that culture is transmitted genetically leads to some philosophically peculiar and borderline racist contortions about the influence of "blood" on a person's character. These beliefs about the primacy of race on a person's character were promulgated most heinously by the Nazis with the Nuremberg Race Laws.
It's unfortunate that the grandmother cannot bond with the child because of his skin colour and, it goes without saying, racist. But have you considered that this racism is an authentic expression of her culture, and that you have no right to criticise such culturally-derived prejudice ?
Other points:
Have you ever visited Senegal, or do you plan to ? It's impossible, in fact absurd, to try to enculturate your child as Senegalese if you have not seen the country.
There is more to a culture than cuisine. How do you feel about the jaw-droppingly misogynistic aspects of Senegal's Islamist culture, not least its leading position in Africa as a practitioner of Female Genital Mutilation ? If your child was a girl, how would you explain that to her ?
You say "everything is so westernised" - which is not surprising because you are in Europe. And "white perspective". Skin does not have a perspective,
The only effective way to inculcate your child in Senegalese culture would be to move there for a year or two, enrol him in a local school, immerse him in language classes, and encourage him to hang out with his peers and do whatever activities they do.
Otherwise, let it go. You will only confuse him for no good reason, unless you consider assuaging your own unarticulated and non-specific "white guilt" about I don't know what, a good justification.