So, how exactly does the peat lower the GH and KH?
Peat moss is variable. Some acts a lot 'stronger' than others.
Peat moss can act like an ion exchange water softener. Peat accepts positive ions like Ca++ and Mg++, and gives off H+.
By the very definition of pH, rising H+ means lower pH value.
Peat moss also gives off organic acids like tannic acid.
You can try getting some of the locally available peat moss and seeing if it helps. I get Canadian Sphagnum Peat Moss in garden stores. No surfactants, no fertilizers. Pure peat moss.
When I start with water that is already fairly soft then the peat moss can really help make it more acidic (lower pH), but IME the material I get does not seem to alter the GH or KH by much.
I have seen threads from some people with a more 'active' peat moss and the GH and KH values came way down for them.
So here is what I would do to keep the really soft water fish:
1) Go get a gallon of reverse osmosis or distilled water. Run some tap water into a bucket and aerate it for 24-48 hours with a small pump. (Air or water pump- just keep the water moving) This ought to drive off the CO2 faster, so the pH will stabilize.
2) Make some blends of aerated tap + RO. See what the GH, KH, TDS and pH are. As hard as your tap water is I think you will end up with a blend of as little as 5-10% tap + 90-95% RO. Perhaps set up tests for 5%, 10%, 15%, 20% and 25% tap.
3) Starting with whichever is closest to what your fish want (research the fish), add a handful of peat moss and stir it around. Test again the next day, and every few days for a week to see if it is stable.
4) Once you find a recipe that works then you will need to make all the water for your aquarium water changes that way. Top off with RO. I keep some peat moss in the filter for fish from black water streams.
5) When you buy new fish that have been kept in the locally hard water you will need to acclimate them very carefully to your soft water. I set up a quarantine tank that matches the water they are used to, and during the month they are in quarantine I slowly alter the water to match the main tank. This will be a big drop in mineral levels for the fish, and cannot be done in just a couple of hours of drip acclimating.