Hmmm.. an IT Flotilla without an Admiral.. Without your own network.. Sounds dodgy.
Does you neighbour run an Apple Router.. that is your one and only chance to get this working, EASILY!! If he does then the TC can work as a wifi extender and all will be good. Apple does not mix with the riff raff like every other router manufacturer.. the very thought.. would turn your Apple bad.
Or perhaps you can buy an airport express and get permission from the neighbour to plug it into his non-apple router and you can connect to that. Otherwise this is just too messy.
You could run WDS to an identical router as your neighbour uses.. as a wireless bridge. Then plug the TC into that and redistribute it in your house.. the neighbour would have to agree as some setup is required.. and the equipment needs to be up to date and capable of WDS bridging.