Sheldon Brown's website has a table that lists cable pull data for most modern IGH shifters.
What this data won't tell you absolutely is where the tolerances lay for slightly different cable pull among shifters still being able to work.
Sturmey 3-speed shifters are known to work with SRAM/Sachs hubs, and there's a modern trigger-style Sturmey 3-speed shifter out there, so that may be your answer.
No, derailleur shifters can't be freely appropriated on IGHs of the same number of speeds. IGHs use variable amount of cable pull per click, and rear derailleur shifters are the same per click. As for front derailleur shifters, I just measured a Shimano SL-MC18 I had laying around at 13mm for the first click and 7 mm for the second, which is probably more than is useful for three-speed hubs. There are variations (Shimano road is some amount less) but most flat bar triple shifters will be in this ballpark. (7mm is actually kind of close to the 6mm your second click needs. Maybe it would work great to put this shifter on a SRAM 3-speed and set it up so the first click simply takes up 5-6mm of slack, I'm not sure about that.)
The nature of 3-speed hubs in particular is that they're somewhat more accepting of a technically wrong cable pull off the shifter. They all want zero to negligible cable pull in the first gear, which means the cable can be slack if needed, and in the second and third gears there is usually a small range of cable pull that can work. Once you have more than 3 speeds, the tolerances stack up such that you don't have this leeway.
The hack you link to is actually pretty smart. The Sheldon data shows that the SRAM i-motion 3 shifters want about 7.7mm of pull on the first click and another 6.1mm for the second. The 9-speed X7 shifter they're using pulls 4mm per click. So by grinding down the first detent they're making a cable pull of 8mm exactly, and apparently they've found they can fudge the cable tension enough to get it to run in the third gear on the next click. (Note that one doesn't have carte blanche to assume that if it gets into the gear then all is well, because it could seem like it's in gear but be partially engaged and wear prematurely or have other problems, but in practice when messing around with internal hubs it's often possible to simply feel where these points are). Personally I wouldn't hesitate to give this plan a shot if a trigger shifter is what you want. The Sturmey 3 trigger shifter option is also a cable pull fudge of sorts as you can see in the data, and the quality is so-so, being a Sunrace-era S/A part. What I would look askance at on that page is the statement that you could do this with any shifter. The X-series SRAM shifters are way easier to disassemble and reassemble like this than, for example, any Shimano trigger shifter.
SRAM never made trigger shifters for their 3-speed hubs and are now out of the IGH business globally.
Another option is to use one of the electronic servo spooling type shifters, i.e. Archer D1X. In theory these are all able to perfectly emulate the stock shifter.