Electronic – use a buck regulator for mains voltage if I put a resistor before it

Your 12V linear regulator is most likely running out of headroom, due to the fact that you have a fixed turns-ratio step-down transformer as its input source (which is of course gets rectified and smoothed). As the AC voltage coming in drops, so must the AC voltage coming out of it. Once the voltage gets too low, your linear regulators don't stand a chance.

You can either replace the whole power supply with a switcher as the other answer suggested, or use a switcher to generate a regulated 15VDC bus to feed the regulators (instead of the transformer and rectifier scheme). If your loads are noise sensitive, keeping the linear regulators may give better performance (at the cost of lower efficiency).

A switching regulator like John suggests would work in principle, and any experienced electronic engineer would probably find it trivial, but keep in mind that that chip is effectively funneling all of that power through a 4mm square plastic package with no leads and that pretty much all the heatsinking is on the bottom (board side) of the chip. This makes the PCB design a bit tricky in that area for the uninitiated. See section 10 of its datasheet.

A less demanding solution in terms of heat removal would be to separate out some of the functions that that chip does, thus providing more surface area for the heat to dissipate, but it only shifts the PCB difficulties into avoiding parasitic interactions between parts that should not interact.

To keep it simple, I'd be tempted to go with a separate, lower voltage transformer/rectifier/filter for the smaller amps. If you happen to find a transformer that has a single primary and multiple secondaries that collectively meet your requirements, you can make it that much more compact, but those are comparatively rare.