Hi Peter!
I like that your design fits in a D-Sub shell!
If you manage to solder the exposed pad with your infrared station and if the RP2040 is properly aligned, then it is quite easy to solder the pins from the side.
I usually use a soldering iron with a chisel tip or hollow tip. Just solder the pins with plenty of solder. Even if you have some bridges or shorts, that does not matter.
Now comes the magic..! Apply a decent amount of flux across all the pins. Not the stuff you get from Aliexpress, but flux from Amtech. My favourite one is NC-559-V2-TF (or V3) Tacky Flux.
Next slide your tip across all pins on one side in one single motion. Magically all shorts are gone with the access solder sticking to the tip of the iron. Clean off and repeat on the other three sides. If you still have some shorts, re-apply flux and repeat. Clean off with some IPA (not the Indian Pale Ale

) and voilá you're done.
PCBs with a HAL (hot air level) finish don't have enough solder to connect to the exposed pad. But you can do the same trick here. Add a bit of solder to the exposed pad on the PCB.
Apply some Tacky Flux to it and place the QFN on top. I usually place it slightly skewed on purpose. Then I heat the board with a hot plate or hot air gun (but it should work with your infrared station as well).
When the solder starts to melt, you can see the QFN slide in place when the solder "grabs" the exposed pad off the chip. Let it cool and solder the pins as described above.
Sounds complicated, but it is not to hard once you get the hang of it. Amtech flux is your best friend
The USB mass storage is fortunately pre-programmed on the chip. I was a bit worried about that as well... But after connecting the first time it immediately showed up as a drive with a text file in it with the board loader version. Once programmed you can force it back into mass storage mode by holding the reset button while connecting USB. Same way as the regular Pi Zero or Pico.