I had the same problem with you and I ended buying a freezer. I bought a small Zanussi freezer from Fortress. I know you said that you don't have freezer space but do you have a place to put a small freezer? You can put it anywhere as it will only be a temporary solution; the living room, the bedroom, anywhere where you can find a small footprint. It's worth it because you can store tons of milk in there. And because it's a freezer, the temperature is kept at -17 degrees C; you can store the breastmilk up to six months. When you're done using it, you can sell it.
There is a place where you can rent freezers but I found that it would actually cost more (about $300 a month, vs. a new freezer for $1,200) to rent. This is a couple of years ago.
Good luck!