Trial and disaster would be a very accurate description. Carbonating with dry ice is wildly inefficient. The rate of sublimation (off-gassing) is much greater than the rate of dissolution (carbonating) into the liquid, so you waste a lot of CO2 to the air. A general rule of thumb for carbonating using dry ice is to use 1lb per gallon. You can normally get between 5 and 7 adequately carbonated 5-gal kegs of soda with a 5lb tank of CO2 by force carbonating in a keg. I'll let you do the math on that one.
Even with measuring out the correct weight of dry ice based on the volumes of CO2 that you want to carbonate to, putting that in a sealed bottle will likely build up too much pressure and explode before it can be fully dissolved.
We had a guy at our church do that with root beer at a Halloween party this year. He had a modified Sanke keg with a wide mouth threaded lid that he dumped root beer mix and dry ice into. The lid had some relief valves and about 10 seconds after he sealed it up, the relief valve opened and blasted the ceiling with root beer and covered the entire kitchen in a fine mist of root beer scented stickiness.
While carbonating with dry ice may be fun, there are better ways to add bubbles to your beverage.