Here is a helpful Deskthority post from NathanA on the subject of ps/2 compatibility:
PS/2 support for xwhatsit controller:
I read in the AT90USB162 datasheet “USB pad multiplexed with PS/2 peripheral for single cable capability” – it would be cool if it had ps/2 support! Would that be possible to do with one of those passive USB-PS2 adapters that used to come with some computer mice, even if the controller has a USB-C connector?
How easy would it be to adjust the current controller to make a dozen or so ones with this chip instead of the ATMEGA32U2 chip? Per pandrew: this chip has “16k of flash memory. Qmk firmware currently uses around 22k. That’s not to say it’s impossible. It’s probably possible to strip away many QMK features and get it working, but it won’t be great.”
I think the tradeoff in some additional functionality would be worth it to the folks who strongly prefer PS/2 connectivity.
I actually looked into this a while back, because I was interested in making it possible to use my F77 with older PCs. My question was whether it was possible to “bit-bang” the PS/2 protocol on the USB pins of the ATmega##uX MCUs to do exactly this (use a passive USB-to-PS/2 adapter).
The short version is that, you and pandrew are correct: it is not possible to do this with the AVR variants that have built-in USB. It is possible to do with AT90USB162. HOWEVER: the big problem is that QMK has zero support for acting as a PS/2 keyboard. It can support adapting a PS/2 mouse to USB for (apparently) certain USB keyboards that have a PS/2 mouse port on them, but cannot directly present a PS/2 device on the PS/2 bus.
The solution is likely to use a different open source keyboard firmware project called ‘ps2avr’ (https://sourceforge.net/projects/ps2avr/) which is made to do exactly this: make non-USB AVR variants talk PS/2 protocol to a host. This means that you would get none of the QMK features though, and the keyboard would ONLY support PS/2, not both PS/2 and USB simultaneously…though in theory you should be able to have a ps2avr firmware for PS/2 support and a separate (stripped down) QMK firmware for USB support, and then flash from ps2avr to QMK if you need to use it as a USB keyboard (and vice-versa when you want to go back to PS/2). Also, either pandrew or someone else would also have to take the time to port the pandrew capsense stuff from QMK to ps2avr, and I don’t know what the relative difficulty level of that project would be.
I believe the AT90USB162 is basically pin-compatible with ATmega32u2, so it shouldn’t require too many changes to the existing PCB design to change the MCU. But yes, it is unfortunate that it has half the memory of the 32u2. I believe there are non-USB versions of ATmega that have the same 32K of memory (e.g. ATmega32A) that would also be compatible with ps2avr firmware, but those are not pin-compatible with the 32u2, so the PCB would likely require some adjustments to the design to accommodate one of those.
For my own uses, I ended up buying an active USB to PS/2 adapter. Yes they exist, but there aren’t many of them, they are expensive, they are hard to find, and they aren’t always compatible with 100% of USB keyboards or 100% of PS/2 hosts.
The one I bought was this one, but when I bought it, it was “only” $75, and it looks like the price has gone up since then. 
I can confirm that it works with my F77 running QMK…but ONLY as long as I have a QMK firmware flashed to my keyboard that was built WITHOUT the MouseKeys feature included. Something about the QMK MouseKeys feature really confuses this adapter (probably something to do with the particulars of how QMK is presenting both a keyboard and a mouse simultaneously over the USB bus).