I've tried doing exactly what you describe, without using counter-pressure to fill the growlers. While you can carbonate the beer just fine, at room temperature you'll get so much foaming during transfer (no matter how long your lines are) that it'll probably end up as flat as it was before carbonation by the time you're drinking it. You could try it with counter-pressure, but a new CP filler can be $50+ (you can make quite workable ones yourself for a lot less I'm sure), not ideal when you're trying not to spend money. Also, at room temperature the equilibrium pressure for carbonated beer is high enough (25+ PSI easily) that it may pose a problem for some growlers, which might not withstand this high a pressure while CP filling.
If you are really tied to the idea, you should at least try to chill the keg as much as possible right before filling growlers, to help maximize the solubility of the CO2, which should help to decrease loss as the beer experiences turbulence and drop in pressure (I used to leave my keg outside overnight). Also, try to push from the keg using very little pressure (or even better, just by using gravity), to minimize turbulence as the beer exits the keg. Again, releasing pressure at room temperature will cause the beer to foam quite a bit, even in the keg, but this will be mitigated by cooling it as much as you can. You could try to dispense through a very long and narrow line (as jsled suggests), or through a pigtail, which would allow you to keep more pressure on the keg, but I can say from my own experience that if you do this without chilling you'll still have a lot of foaming going into growlers, since any way you cut it you're still going from high to low pressure.
There are devices that exist for serving room temperature, pressurized beer by passing it through a heat exchanger, or cold-plate, that is kept in a bath of ice-water. Obviously they're fairly expensive, but you might be able to build one yourself by making a coil of tubing, submerging it in ice water and passing the beer through it on the way to the growler, to flash-cool it. Could be worthwhile especially if you were to bang out a whole keg's worth of growlers at once.
Anyway, good luck with this, hope you can figure something out. The unfortunate thing is that you'll probably end up spending money to make this work and at a certain point it might be just as easy to shell out for the fridge, which will probably make more sense in the long run.