I ordered in some replacement parts for both of the issues mentioned in the previous log.

First, I replaced the relay with a darlington pair BJT. This offers absurdly high current gains for very low Base currents. In other words, it will work to power my windshield washer motor, which needs around 2.5A to function, with the < 16mA I get from the RPi gpio. In the attached breadboard picture, I have a test setup for this.

Second, I replaced the 5V LDO with a 5V switching regulator. I ordered two just to be safe (lm2576-5.0WT and bd9874cp-v532), which also allows me the flexibility of putting these regulators in series if they still get too hot. After leaving this circuit to run for a few minutes, there was virtually no heat coming from the regulator. We'll count that as a win!

As I mentioned earlier, the attached picture is from the component testing I did on my breadboard. After confirming these components are good solutions, I went back and did my board design for v3 of this project. My schematic and Layout are now as follows.