It sounded to me like the primary issue might be obtaining the material to compost rather than how to compost it. I myself do not compost as it is often thought of. I don't worry over proper mixture of green/dry or whether or not a proper temperature is achieved to make it biologically sterile. I just put stuff in piles or use it as mulch until it rots.
I'm also not accustomed to gardening in an urban environment, but I think gardening as much as you can, wherever you are, is really important. To get stuff to compost maybe you can scrounge it in the neighborhoods or in a rural area if one isn't too far off, and you have a way to collect and haul it. If your neighborhood or one close by has trees you might find people who would let you rake and take leaves or just swipe them already bagged up by the curb. A simple rule to me of what I might call "natural" gardening because I don't like any of the other words people use is just don't poison it. With that in mind try to make sure you don't collect stuff from a "chem lawn".
If you do have access to a rural area lots of good stuff often accumulates in roadside ditches, especially if the area is wooded. You might be able to jump out of your car with a big trash bag and swipe some. I'm not all that sold on the whole Hügelkultur thing especially if you don't really have space for it, but rotted wood is great to compost, bury in the ground or put in the bottom of containers. It needs to be well-rotted as in you can tear it up with your hands or it will take too long to break down, it works kind of like a sponge too. You might could even swipe that from a public park or something. You're allowed to collect and burn downed wood in campgrounds so why not just bag up some of the more rotted stuff and take it home.
I think closed loop system in a very small space might be tough, but I think the budget issue might be solvable and without wondering just exactly what might be in that purchased stuff.
