Oh boy!
Tricky! And the gap between her kids is close so I'm imagining its definately attention because he probably noticed all of a sudden new cooooo baby got attention! If you see what I'm saying! Thing is its not good if hes like that with 'Mum' too

Has his Mum spoke to a GP about her sons outbursts I mean there 'can' be reasons from behaviour problems to hyper-active I think. No experience on the matter personally just some thoughts and/or possibilities.
I often think that other people should not tell or tap (no not smack just tap) the hand of another persons child unless told they can do so or a relative but I think there comes a point where you should be the one to say 'stop' in a deep NO tone & I think this would be one time - if you don't already.
I would also suggest to the Mum that son has Mum & me time - maybe a weekend where they can go do something together & Dad has youngest!