I got new parts from Mouser and finished populating my boards. First power up didn't work completely: 1.) the LEDs blinked so fast they looked like they were always on and 2.) most of the time the display wouldn't turn on at all and I'd have to restart the machine. I found out the problems were caused by 1.) no delay in my pattern display demo program so the characters cycled so quickly, every segment looked like it was on. 2.) I soldered the wrong timing components for the initial condition setting circuit. Instead of 10kΩ and 22nF I needed 100kΩ and 100nF. I used these correct values during my breadboard test - I'm not sure why I put the wrong values in the schematic ¯\_(ツ)_/¯. These simple fixes got it up and working nicely.
I updated the firmware on the Arduino to read out data from the sensors and show it on the display assembly. I noticed that I flipped the enable wires for the 1st and 3rd characters but is fixed very easily in software. Temperature (C), relative humidity (%), and pressure (kPa) are displayed.