My tank holds 4 gallons. It is a low-flow installed in ‘85. It has no mineral buildup after 27 years. It has fairly new guts that work perfectly. It takes at least 2 flushes to do the job and has since day one. The only design that would change the equation is one that used more water.
We just remodeled our baths in San Diego. The low flow toilets we installed work fine..first time, but they now have larger flush valves. Same amount..faster flush.
We also installed a on demand hot water circulating pump..we push a button and it circulates the hot water to get hot before you turn it on. These things have cut our water bill by 30%. The pump cost $200 and I installed it myself under the sink.
I would tell my customer the simple fact, they need a new toilet.
You got one of those low flow toilets during the period that they were 3.5 gallons, and many of them were lousy, they went through the same process as with the 1.6 gallon flushers (from junk to OK).
At first the manufacturers just used less water with the old designs and bowls, they did that with the early 3.5 gallon water savers and then these 1.6 gallon water savers, but the problems have been worked out now, do some research and you can find great toilets to replace that faulty one of yours.
Personally, I use old 1.6 gallon toilets that I have removed from customer's house because they were weak models (but attractive, one Kohler, one American Standard), and they work fine for me but I do keep a plunger close if they need a bump. If they were an issue, I would go buy a new Toto, or whichever brand is a good one currently.