As many piccolo players know, tuning can be a wreck on piccolos. I play piccolo just for my high school’s marching band and I’ve always have had tuning problems even when I got a new one. I thought it was the piccolo but even my flute teacher couldn’t figure out what was wrong. She watched me play and said my embouchure didn’t change when I switched octaves. I would tune to A, per usual, and it would be in tune. Then I play a middle register F and it goes so flat on the tuner that it’s the furthest away from being in tune. My F#, D, E, etc. are all the same. And everything from around C above the staff to the highest extent is fine. So then I tuned to F. All of the notes I was previously having trouble with were in tune but now A and all the others are sharp. Not as drastically as F but I still have to roll in a bit. And of course with marching band and weather, this can be an inconvenience. I have yet to come across someone who has had the same problems as I have and I’m really concerned. I highly doubt it’s my piccolo considering it’s only about a year old and kept in good condition. I also got it recently touched up at the repair shop. I had these same problems on my old piccolo too. So it must be me but my flute teacher can’t identify anything and I don’t know what to do.
It sounds like your flute teacher is separate from your band teacher. So, consult your band teacher about what's going wrong tuning-wise. Make sure to mention that you had the same problems with your old piccolo. Your band teacher may have more insight about how to fix the issue.
If it turns out that your band teacher isn't experienced enough with the piccolo to help you, or if I'm wrong and your flute and band teachers are one and the same, see if you can consult another piccolo teacher's opinion. If necessary, you may need to switch flute/piccolo teachers.
It sounds like the scale is a bit stretched; do I understand correctly? That is, when you tune F to the tuner, A comes out sharp, and when you tune A to the tuner, F comes out flat. That means that there is too much distance between your A and your F.
But you should worry less about the tuner than about the other players in your band. If the band is careful about tuning, you'll want to play pitches in slightly different places depending on the harmonic context. For example, if you're playing your F against a D-flat in the tubas, it might be perfectly fine for it to be flat.
See, for example, Taming the Little Beast: An Introduction to the Piccolo, which I found with a cursory search. It notes that you should expect some notes to be slightly sharp or flat in any event, that high notes on the piano are intentionally sharp, and that major thirds should be about 14 cents flat compared to equal-tempered thirds. It also has some nice additional pointers
There's more to staying in tune than embouchure. There's the whole breath control thing, and to some extent the amount you "roll" the mouthpiece to adjust each note.
It sounds like your flute teacher is not competent. If she hasn't even tried playing your instrument herself to verify the intonation, then she's not helping at all.