Can’t comment on your chances of getting into Dulwich at 7+ with state + tutoring, but we are parents at EHTM and are very happy with the school. It seems to be a really nice mix of learning in a fun environment, doesn’t come across as too pushy, and they made some recent improvements to facilities etc. We’ve been impressed with their recent response to the pandemic as well - home schooling was tough but really well supported by the school and they’ve made things as normal as possible this term.
I hear good things about Broomwood as well, so really depends on the feel and the culture and whether it is right for you. I think Broomwood is perhaps more ‘English’ whereas EHTM is quite international.