The voltage regulator worked.
First order of business was to adjust it. I hooked the input wires of the regulator to the battery terminals and started the engine. I adjusted the pot until I got 12.6V. The adjustment is quite good, I'll say. I chose 12.5V as that is a pretty standard voltage you would see on a battery without the engine running. Time to try.
I first placed the regulator on the battery wires feeding the DRL controller. Much to my dismay, the flicker persisted.
I was feeling froggy, however, so I decided to place it inline with the turn signal trigger wire. Success. No flickering.
The downside to this is that it would require two regulators - one for each turn signal. I concede this is not very elegant. Twice as hard to hide and twice as expensive.
Not satisfied to leave it there, I decided to take some measurements. I first measured the voltage on the turn signal wire when blinking. I'm going with 14.3V. I then placed my DMM in ammeter mode and measured the current on the turn signal wire feeding the controller. I came up with 12mA. Using Ohm's Law, this means you could achieve the same voltage reduction to the controller using a simple 150Ω resistor! Additionally, these values mean the resistor will only be dissipating a measly 22mW of power, so you can use pretty much any resistor, including small/discreet/inexpensive resistors. And if you're using something like a 1/4 W resistor, I'm sure you could cover the resistor in heat shrink without any ill effects for a very sleek inline appearance.
I have a box of resistors laying around here somewhere but I'll be damned if I can find them. Oh well, a nice 1,050 piece kit of resistors (38 different values) on Amazon is $10.50 with tax. These are 1/4 watt and have a ±1% tolerance. Added bonus, if my measurements are off, I have other resistor values to experiment with. I have purchased it and will experiment again when it is delivered on Monday. If it works, I think you will be hard pressed to finding a cheaper, easier solution.