It's better to have the tune for stock exhaust and add PCIII, than to have the tune for the tors+by-pass and add PCIII, is that correct?
This is complicated. If you're using a stock tune, the secondary plates will be partially closed as a function of gear and rpm. This means less air gets in. A PCIII can't change that. There are only 2 ways to open the secondaries: 1) take them out; 2) Tuneboy.
If you take them out, you can add a PCIII to increase the fuel. But there's still a problem with the underlying Triumph tune. According to the tunes on the Tuneboy web site, the Triumph tune for stock exhaust and cat (20222) closes the secondaries by different percentages as a function of rpm but also by gear.
However, the stock Triumph tune for TOR's/cat bypass closes the secondaries by different percentages as a function of RPM but it's the same in every gear.
Why is this important? Let's say you've removed the secondary plates and install a PCIII. The PCIII dyno tune is done in a single gear, usually 4th with fuel adjustments made as a function of throttle position and rpm. if your base Triumph tune has the secondaries closed by 50% in 4th, it is supplying fuel for that percentage. The ECU doesn't know you've removed the secondaries but the PCIII will add the amount of fuel needed to match the extra air flow. But if your base tune closes the secondaries by 30% in 3rd, there's no way you'll get the right air/fuel mixture in 3rd gear by tuning the PCIII in 4th.
Bottom line is that if you go the PCIII route, make sure you have the 20226 tune loaded which closes the secondaries the same in all gears.
Wayne at Tuneboy pointed this problem out years ago.
Second bottom line is that if you have Tuneboy, you don't have to worry about this crap. Tuneboy has custom dyno tunes for most intake and exhaust configurations. You can always add a PCIII to optimize your tune even further. If you can borrow or rent a PCIII, you can generate an optimal trim map with a dyno tune, apply the trim to your Tuneboy tune, and then remove the PCIII - or leave it in.
Bottom bottom line is that there's no point in using a PCIII unless the secondaries are removed or opened 100% by Tuneboy.
Easy, huh?