Neighbourly relations are poor so can't see this working!The other slightly more expensive option is to approach the neighbour and suggest removing the beam entirely and replacing with steel higher up, your neighbour may appreciate the extra headroom too, the question is whether they want to share any of the costs. You could offer to pay for calcs, BC and insurance, if they go halves on materials and labour. Can't hurt to ask?
Thanks to all feedback and suggestions. I'm now seeking second opinions.

