I think you'd be OK to put your savegame in your new Rome\Saves folder - as long as the patch version of the savegame & your new install is the same!
I move most recent save back & forth between 2 computers all the time on a USB memory key with no trouble.
If you otherwise want to just unlock all normally playable factions as fast as possible without tweaking the files, just win the Prologue. Set difficulty settings to whatever suits you & beat the Greeks on the special Prologue mini-map. Takes just a few hours.