If the software can tell a difference between APs, then the APs are not working as they were designed to work. Theoretically, if the firmware in the WiFi module were capable of being placed in promiscuous mode, and that were done, and the network stack were to selectively send raw packets to specific MAC addresses, you could pick a particular AP to send packets to.
Most likely the have a bad router. The best way to fix this is to get them to configure it correctly if it's a software issue or to replace it/just turn it off if it's a hardware issue.
Consider that if you were able to change settings in such detail, you could work around the problem for you, but all of the other library patrons would still have problems. The best outcome for everyone is to fix the problem so that everyone gets good WiFi access, not just you.