A bridge rectifier has two diodes in series with the load, and even for Skottky's 500mV drop is quite low at 15Arms current (and maybe 45A peak current). You can add a shitload of bulk capacitors. this will reduce ripple, but at the cost of ever growing peak currents, and you may saturate the transformer during those peak currents.
Ap bap bap*--- saturation is a magnetization thing, not a load current thing. Load current doesn't affect primary voltage (at least, not in an unbalanced way, given the full wave rectifier). Indeed, flux density
decreases as load current increases, because of primary DCR. Which is a small effect, for the most part.
*Um, hmm, I have no idea if there's a preferred transcription for that... it's a, not really a turn of phrase, more a sound. I imagine it's a simplification of something like "ah, but but--". ...Things I'd never have thought about, but for foreigners/ESLs being present..!
Also regarding transformer impedance / regulation / efficiency, toroids are generally quite good at that, so even if it's a small transformer (and it sounds like it's not), I'd expect pretty good regulation, even while using large filter capacitors. Again, assuming you don't mind the low power factor.
It's not saturation, but overload problems I think, I'm not too well versed in transformer details.
Yup, just RMS current to worry about here

1. Use the voltage doubler (Only one diode in series so already less loss).
2. Use a moderate capacitance.
3. Add an SMPS circuit to get 12Vdc.
It's higher efficiency both for the SMPS and the more even load for the transformer, and you avoid problems with headroom.
Note that doublers draw worse PF as well; at least for the same output ripple. Though if capacitance should be "moderate", that should work out favorably overall? For certain values of "moderate". I see your point. It's not like the full doubling is needed; a lot more ripple means higher PF and (somewhat) lower DC out. Which, being that DC would be "too high" here, that's advantageous.
I think at that point, I'd be more worried about the capacitor(s) themselves -- electrolytics even at line frequency tend to be rather unhappy with more than say 10% ripple. Polymers might handle it fine? They're rather pricey though, for the kind of values needed here, nevermind film or ceramic...
SMPS can also be a SEPIC type, allowing boost or buck output (freely and continuously above or below input) -- but any SMPS is probably a bit more advanced than OP is looking for here, unfortunately.
There's also the boost/buck (transformer) strategy, where you take a low voltage transformer in series with the main winding to make a smaller change to it. Well, in this situation, good luck finding say a 2Vrms transformer, right? -- but, being it's a toroidal, there might be some room to add windings around the thing by hand. Hacky, but it's probably in the couple-turns-per-volt range, so, I'd guess less than a dozen turns would do, and, just use whatever hookup wire is rated for the (RMS) current required. Does make mounting it a bit annoying -- avoid crushing the added wires.
Tim