Best case would be to put the peat in the NEW water. If you put it in the tank you're just setting yourself up for a bigger swing.
Really, though, you should test your water out of the tap first (let it sit overnight before doing so to let any gasses that may affect your reading escape). Then once you have a reading, you can figure out where to go from there.
Keep in mind most CA and African cichlids (and some SAs) do great in hard water. Also, many that don't have been farm raised for so many generations that they do fine too.
If you do find out your tap is much too hard for your situation, just make smaller changes more frequently or treat it (peat or an R.O filter) prior to adding.