I got two setups similar to that (using mAP Lite), also for printer or PC/printer (with stupid hub in between).
And one of them is giving me the most trouble as far as printer issues are concerned ... (max 3 times per month, so still not thát bad)
One for a PC needing connection to internet and having a dedicated printer. Source of wifi is other brand, so I configured that mAP Lite in station mode, local router functions. All done.
Second one is from Tik to Tik with printer needed to be accessible from network through wifi bridge.
This one gives me trouble from time to time.
No way to have a dedicated cable (not even powerline solution) for that location.
First scenario:
when wlan2 is configured as station, it should not be part of the bridge or it's DHCP client will not work (that's why you don't see it from the router).
On that bridge you need to foresee yourself a separate subnet with DHCP server, NAT, other router functions etc. for clients connecting to ether ports of hap. Usually setting the device in AP mode should foresee all that ?
From your router you will not see devices behind hap, but devices behind hap will be able to contact router.
Unless you want to be able to control that printer from the other side of hap ? Depending on the source of your wifi (Mikrotik or other brand) it may work (if Tik, but then you need to move to station-bridge), or not (other brands).
Which brings us to ...
Second scenario:
Are the packet losses a problem for the PC and printer ? Usually TCP/IP will retry in case of drops.
Is the AP you're connecting to also Mikrotik (use station-bridge then) ?
Can you clarify more on the purpose of that PC/printer combo ? Is it standalone or does it need to be controlled from the other side of hAP ?
What is the source of wifi where hAP connects to ?