First of all, it´s correct, you can´t save replays of campaign battles, only of custom battles.
If you want to play the battle replay as it´s saved by RTW, then, of course you need RTW installed, because you als get the whole thing like camera control in the replays (you can watcht the battle from any angle you want, not what you saw while playing it), that, as you can guess, require the RTW software to work.

Now, the alternative would be to do the battle in RTW, save the replay and then use a screencaption/movierecording program to make it into a video file. Those however are sort of WYSIWYG programs, means they record what´s shown on the screen, so making a decent movie requires some editing.