Great!
Unfortunately your call sign must get sent both ways. There is no legal way around this. Just like any broadcast.
You could always create a vanity npub for HAMSTR only or something as well
Only the server sees the npub, over the air it is compressed and split into packets, so while technically possible to sniff and decode, unless you knew it was a HAMSTR signal and had the proper encoding type, there is not really any way to get at the info. Possible? Absolutely. Likely? No.