The *1084 linear regulators have a dropout voltage of 1.3v at 5A (they're made by various) and they're not that expensive. At less than 5A the dropout voltage drops a bit.
At less maximum current (let's say 3A) you have quite a few options (more models of linear regulators with dropout voltage below 1v) ..
for example SC1592, LT1764, MIC29300 to MIC29303 (some models may be 16v max input voltage) , these are all with dropout voltage below 0.5v
for less than 1v you have for example LT1185 with about 0.7v dropout voltage
and i've only looked at Farnell in the LDO section, Digikey has other manufacturers and more chips to choose from
Anyway, the point is that most of these linear regulators can be connected in parallel, with a suitable resistor between their outputs to balance them ( in
the datasheet for LT108* at page 15, Linear recommends using a meter or so of AWG #18 wire as resistor, that's about 0.02 ohms - instead of wire you could just use a proper resistor).
So if you can't find a cheap 5A linear regulator with less than 1v dropout voltage, you could opt for two 3A linear regulators in parallel that work with less than 1v dropout voltage.
I've looked through all your posts and you don't seem to post exactly how much current you want... so I can only give you general advice.
also.. for 3-5A of current you may be able to find separate Schottky diodes with low forward voltages in the range of 0.3-0.5v, so the voltage drop on a custom bridge rectifier could be less than 1.6v
ex
0.8 uk pounds each
VISHAY VBT4045BP-E3/8W has ~ 0.3v forward voltage at 10A , ~ 0.4v at 20A
0.7 uk pounds each
STPS20120D Schottky Rectifier, 120 V, 20 A, will give you ~ 0.6v at 10a, about 0.5v at around 3A
0.38 uk pounds
NRVTS1545EMFST1G Schottky Rectifier, 45 V, 15 A, less than 0.4v at 10a (cheap but a bit harder to solder and would be harder to dissipate heat from them, compared to to-220 models that could be put on heatsink.
and many others.
Don't look just at diodes rated for peak current a bit higher than your maximum, and don't filter results based on the forward voltage.. because for example you may find a diode with forward voltage of 0.8v at 25-40A but the forward voltage would be much lower at 0.4-0.5v or even lower at 5-10A you may have.
You have a good example above, where the 40A diode is listed with ~ 0.67v forward voltage drop on the site, but in datasheet if you check the graph you can see it has much less, about 0.3v at 10A.
So if I had filtered to see only diodes with forward voltage less than 0.5v I would have missed this one.
also another note... if you could go with switching power supplies, there's also laptop adapters that output 12v (these are usually rated for 65w) and there's also adapters like xbox power supplies which output 12v and something like 120-160 watts (i don't know the exact figure).
Laptop adapters have the advantage of simplicity, passive cooling, already having wires and end connector that's convenient, no need to short ps-on wire to ground to turn them on , no 5v standby circuit always sucking power etc etc
There's also tiny ac-dc converters available at electronic distributors like Digikey and Farnell and others which are cheaper than atx power supplies and have either fixed 12v or adjustable output.
ps. The majority of linear regulators won't just quit outputting something if the input voltage drops below the minimum the regulator wants. So you can configure the LDO or LDOs to 12v (or use a fixed output 12v regulator) and if it happens that input voltage is less than desired, the regulators will simply output less than 12v - the regulator will constantly try to reach the configured 12v but will keep working and give your circuit less than 12v.