You only need 'Mechanical Systems' and 'Engineering Mathematics' at level 2 to keep mechanical engineering open. I don't know about physics, but you should only need to do two or three subjects at level 2 for that as well. You sure you can't do 2 breadth in second year?
Yes you can do level 2 management as breadth in first year so long as you've done the prerequisites. I'm an engineering student and I'm doing management for my breadth. Didn't do any last year, but I'm doing Organisational Behaviour as my first, which is a level 2 subject which doesn't have any prerequisites.
Other breadth to consider is a language. I'd recommend a language, personally. Invaluable skill to have, despite how hard it is.