because I believe it can take up to 5 days to get them approved in the US and you can't fly until this happens
I'm pretty sure you didn't intend to write just that, as it's not completely accurate.
If you're a "category 3 alien" you can start flying training once the TSA acknowledges receipt of your fingerprints. The TSA does the background check while you are undertaking flying training and if they find anything amiss they will inform the flight school, which has to stop training you at this time. But in the meantime you are legally allowed to undertake training. If you find a place that can take your fingerprints electronically the receipt acknowledgement usually comes within 24 hours, while prints in ink will take a few days before they arrive at the TSA.
If you're a "category 1" or "category 2" alien you have to wait until the TSA finishes your background check, or until 30 calender days have passed, whichever comes earlier. Only then are you allowed to start your training.
So there's no legal basis for the five-day period that you mentioned. Other than that five days is on the safe side if your prints are made in ink, and need to be fedexed to the TSA.
https://www.flightschoolcandidates.g...§ion=FQ#C4