For the record, I've found that the power supply is providing the correct power to the inverter. The inverter takes the following voltages as per Lutiana of the Vintage-computer forum:
Pin 1 (Grey Wire) - +12V
Pin 2 (Black Wire) - Ground
Pin 3 (Black Wire) - 3.3V for back light On/Off (0v = off; 3.3v = on)
Pin 4 (Black Wire) - 3.3V for back light brightness. I assume the actual volatage equates to the brightness level (ie 0v is 1% brightness and 3.3v is 100% brightness).
These are the values I've read on the inverter. So the inverter is good (and so is the old one).
When the display starts up, there is a momentary flash where the backlight works. I should note that the intensity of the flash is quite dim. I've read that when the inverter starts, it provides a boosted voltage to "ignite" the lamps around 600V. After this brief ignition, the lamps run at a much lower voltage. It's this starting voltage that produces the flash/blink at the begining. Given that it's dim when this happens, tells me the lamps are spent.
It's evident the inverter is getting the proper signal to run the display, but not able to excite the tube any longer.
Just for fun, I will re-measure the signal inputs for the inverter and see if I can get both trigger and intensity pins to read 3.3v.
Replacement lamps are as follows: 2.4mm diamater, (445 - 450mm) in length. The range is there due to variences in measurement methods. There is no part number for the tube as the diamater and general length pretty much defines the voltage etc.
Cost for getting tubes range dramatically. There are pleanty of places willing to charge you $10 per tube with a $30 shipping fee. A great business for these people having a cost <$1. To find a more competative product, look for tubes of 445mm in length. I found an eBay supplier offering 20 tubes for $18 + Free shipping. That's ~90¢ each.
A replacement guide can be found here on the web.