I’m a big fan of 7812 and associated regulators. They are NASA reliable, cheap, and usually stupid simple to wire up and use. Except! There have been counterfeit ones sneaking into the market.
All linear regulators require an input voltage higher than their output voltage. A datasheet somewhere might claim 1.5 volts, I have found that’s not enough. More like 3 (or more) volts if you never want to experience dropout.
Sounds to me like you would want an ordinary 7812 and just insert a diode into the ground leg so that your 12 volts output becomes 12.7 volts. In other words, the ground of the 7812 references .7 volts instead of zero volts. Or two series diodes if you want a 13.4 volt output.
Buy this, when you are done and they work I will answer your question
https://www.amazon.com/Soldering-Practice-DZRCOXI-Training-Soldering-All/dp/B08KDLR6P6/ref=sr_1_10?
“the number of LED’s (actually 3-LED modules) in parallel may vary. (Long story.) So, I don’t want to use a typical constant current LED supply.”
No, you DO want to use a constant current output. Then it doesn’t matter how many LED modules you string together, the brightness will remain constant.
The only thing you have to worry about is if the output voltage from the regulator is high enough.
Measure the voltage drop across one module at the brightness you want, then multiply that by the number of modules you want to string in series. That will tell you how much headroom you need.
A diode is a current device - so look at the LM317 and other methods to make a constant current rather than voltage. Some simple ones are transistor only, there are a few LM317 configurations.
and there is an advantage on premature failure. Basically if one led fails, the constant current is maintained through the other LEDS (so each does not need a resistor). in a voltage circuit, the current is usually set to a value dependent on other devices in the circuit.
I am sure an electronics guy/gal will make what i am saying clearer... but a bit of looking will show you a ton of stuff
forum.digikey.com/t/constant-current-led-lighting-premature-failure/2686
Rather than a voltage regulator have you considered an adjustable Buck Convertor? Both voltage and current are adjustable and remain as a stable output even when the the source input is unstable. We use these on Solar panels that are rated much more in voltage max than what the charge controller is rated to receive.
Look at Texas Instruments’ LM5177 buckboost datasheet. It’s more than uou need, but will work and is auto-qualifird.