Hi,
Have you tried to use a driver for another model included in this universal driver package? Maybe something like the M2020 or M2070? This may or may not work with macOS 13 Ventura (V3.93.06 seems to be for macOS 12 only on HP's support site).
The SCX-3405FW is not mentioned in the December 13, 2022 en-us version of About AirPrint - Apple Support. However, it is interesting to note that a Samsung SCX-3405FW was listed in an old Dec 10, 2015 issue of the same article, as a printer supporting the driverless AirPrint technology. An early AirPrint/firmware version? Anyway, if nothing else works, you could perhaps check whether it would be possible to add the printer (without additional drivers) using AirPrint. That is, if the printer and the Mac computer are connected to the same network (Wi-Fi router). Possibly with updated printer firmware.
AirPrint works with wireless (Wi-Fi) or wired (Ethernet) network connections, so you could choose to wire the printer to one of the LAN ports of the Wi-Fi router, instead of using Wi-Fi for the test. An Ethernet connection is often more stable.
You may have to reset the printing system between tests. See also:
Reset the printing system on your Mac to solve a problem – Apple Support (UK)
Add a printer to your printer list so you can use it on Mac – Apple Support (UK)