There is a community FAQ specifically on locks, I suggest you take a look at that. It also mentioned some specific issues that people have with older Schlage locks.
As far as beaming repeaters, pretty much any mains powered device released from 2020 on will support beaming. If it’s Z wave plus supporting S2 security it almost certainly supports beaming. Most of the other newer devices do as well.
The lock FAQ also explains how to find out whether a particular device supports beaming or not.
FAQ: why would I need another beaming repeater if my zwave lock is already close to my hub?