I live in a similarly v.posh area, now I have to come clean that we have a similar on going debate : about a washing line that would be much more efficent if placed in the front , but according to Mister Sense to have the washing out front 'would be just too pikey for words' . If the trampoline will fit, not be a distraction to motorists, and you don't mind the rows over other kids being unable to resist it, do it. We do have ours placed to the front and side of our house ( so it is obviously not that bad, or Mr Sense would never have placed it there!) It does not overlook anyone, our kids are considerate to others when they use it ( low noise , mainstream hours, etc) and it does not squeak - unlike all the others in the neighbourhood, kids not at security risk -very low foot traffic in my road, and thay have to leave the door open and tell us when they are using it so we can supervise informally.
Kids paraphenalia is not usually considered complaint worthy, unless you have real miserable sods surrounding you. Presumably you have worked hard to live there so - I'm with the missus, just do it!