It's crucial the power from the ATXRaspi to your Pi is supplied from 1) a good power supply, and 2) using good wiring that is either soldered directly (best) or has very good/high quality crimped connectors.
There is very little voltage drop across ATXRaspi itself, but weak connections introduce a lot of resistance between the ATXRaspi output and your Pi, and that's what is causing the tiny voltage drops which the Raspberry Pi (it actually has a voltage monitoring chip) will interpret as low voltage and the Pi will display the rainbow/thunderbolt low voltage warning.
I think in most if not all cases, even this is OK and the Pi will run just fine without any issues, except the annoying icon that will show intermittently, especially when the Pi is busy or working hard.
If you use a USB cable, from the ATXRaspi, you can either add another jumper wire from any of the "+" to any of your Pi 5V on the GPIO header, or only use the GPIO (ideally not both because there is a fuse on the Pi USB but not on the GPIO 5V). Sometimes depending on the total load of your Pi, a USB cable that is not very beefy can also cause the warning (imagine all the extra metal to metal connections that are introduced, versus a straight solid wire - all those are basically resistances that add to the drop and the less connections you have the better).