Eh, this isn't actually that bad. It's quite easy, using a jet aircraft, to (via a ballistic trajectory) get into space. Surrounded by reaction mass as you are, the rocket equation isn't nearly so bad. You could conceivably dock with something in orbit, if you had elastic tethers or something to make the acceleration survivable. If you're much lighter than what you're docking with, you won't knock the satellite out of orbit.

Heck, you could even ignore the jet aspect, and go for full rocket. (An ICBM only weighs around 50 tonnes, after all.) Getting the heavy thing into orbit, though, requires proper, multi-stage rockets.