EDIT: Here is a much better solution I worked out with help from a guy on an automotive forum.

Basically we switched everything to the low side, wired the LED controller positive directly to the battery and the controller ground to the COM terminal of the relay. Then wired the NO side of the relay to the dash switch and the other side of the switch to battery ground. The NC side of the relay is then wired to connect to the ground wire of the dome light circuit. The relay is then triggered with the ACC circuit to switch ground from the dome light circuit while car is off to the dash ground switch when ACC is on.

New video showing updated wiring and effect.

Original answer for context: Thanks for everyone's input. I decided to post my own answer since the solution was a combination of multiple different answers.

Indeed, separating the two circuits with diodes was part of the solution, but finding an adequate signal wire that would either power the underglow directly or trigger a relay for the underglow was the tricky part.

As the dome light behavior is what I wanted to mimic, I tried tapping into that circuit only to find it is a constantly on 12V signal whether the light is on or off, being dimmed and then turned off on the ground side by the multiplex control unit (MCU), producing a temporary PWM signal until off. Tapping into the ground wire between the dome light and the MCU and then grounding that tap elsewhere bypasses the MCU ground and keeps both my tapped circuit and the dome light always on whenever the car is on.

I was able to find a connector in the A-pillar that has a wire that is off when the car is locked, produces a 50mV signal whenever the dome light is triggered by the car unlocking or door opening and ACC still off, and 12V when car is on. I then found a 12V relay with a high and low trigger setting that when set to low, would trigger the relay only when the 50mV signal was present and would go off when either 12V or 0V were supplied, and used that relay to power the underglow from a constant 12V battery source.

Here is a video I made that shows what I did.

The only minor issue I would like to fix now is the quick temporary buzzing of the relay as it turns on and off, as it cycles through that PWM signal, but I'll ask that in a separate question. Thanks again everyone!

Series circuit

^{simulate this circuit – Schematic created using CircuitLab}

Figure 1. Five LEDs in series.

LEDs aren't linear (current is not proportional to voltage as in a resistor) and small increases in voltage can cause a large increase in current to the point of destruction. A series resistor will limit the current to a safe value provided voltage doesn't deviate too much.

Since you require 2 V per LED at 10 mA that means we need to drop another 2 V in the series resistor. From Ohm's Law, \$ V = IR \$, we can calculate \$ R = \frac {V}{I} = \frac {2}{0.01} = 200~\Omega \$. 220 Ω is the next highest standard value.

Parallel connection

^{simulate this circuit}

Figure 2. Parallel circuit.

This time each LED requires its own resistor which needs to drop 10 V across it. Using Ohm's Law again we get \$ R = \frac {V}{I} = \frac {10}{0.01} = 1000~\Omega = 1~k\Omega \$.