Unfortunately, without PCB and a measuring scope, there isn't much that you can play with other than try to match the IC manufacturers recommendations. If your inductors are a good match for the ones in the TI data sheet, you should get a clean output. It is the question if your new source power-supply choke is big enough to isolate switcher noise from that 12v supply going throughout your system. The other problem is that this noise can be down in the low KHz region. Your PCB may be clean, but noise looping though even the ground of your source 12v supply can create a problem in the dreamcast where using a battery or linear 12v DC supply will be completely isolated from your AC wall outlet in a way no cheap Aliexpress 12v AC adapter switcher would be. That is unless you make sure such a 12v switcher came from a reputable design.
(Youtube channel DiodesGoneWild ->
https://www.youtube.be/channel/UCQak2_fXZ_9yXI5vB_Kd54g tears apart many of these switchers, he opens and unwinds the transformers themselves, and except for top name brands and an occasional fluke, most of them are garbage and fail on many levels.)
As for the switcher being 2x2mm. Yes that is a headache, though when ordering PCBs with a SMT stencil, the paste application is easy. Especially on small PCBs. And when placing a device with tweezers, you do need to get it close, but not perfect as the device centers itself as the solder paste melts.
I've seen youtube videos of those using a frying pan with sand to spread the heat evenly before it reaches the PCB, but, the iron has a thermocouple right on its surface with precise temp control so nothing will get burnt.
Having mounting hardware means if you make a mistake or blow the component, or need to change a component, you can remove the part, clean it, and try again. Though, I cannot blame you about wanting to have someone else already equipped and experienced to do it.