the 5.7 litre capacity is 100% fill, you cannot change this from the drain bolt as about half of the ATF is trapped in the torque converter and shuttle pulleys, The person telling you that the transmission will get airlocked is 100% goofing off, the transmission hydraulic circuits cannot airlock.
I have done several hundred transmission fluid flushes using the transmissions own pump and never had a problem. You can be extra safe by doing such flush 1 litre at a time if you are not confident and doing this the first time. Before embarking on this repair first make sure you know how to check level, (meaning see if you can see dipstick and can reach it. Secondly find a thermometer that can be used to measure transmission sump temperature as you would need to check it when hot.
The cooler line procedure only works in transmissions that have a cooler circuit that goes to the radiator, if its like the stupid belta/vitz gearbox where the coolant goes to the transmission the procedure wont work.
btw - before any oil change or flush, first lets see whats up with the error code - you say that you see P1708 suzuki,
This can be quite tricky as its a manufacturer specific code not a generic one, however it is mostly used by others to determine a "switch" malfunction pertaining to either accelerator or "clutching activity" which also traces back to accelerator input problem.
Did you have a problem with the pedal assembly recently?