I had trouble finding the specifications of the bridge rectifier. I believe it is a 1.5 Amp 400V rectifier. Following SeanB's advice I ordered a higher current rated rectifier, 2 Amp 400V:
https://www.digikey.com/product-detail/en/2W04G-E4%2F51/2W04G-E4%2F51GI-ND/604798. I ordered five of them so I can replace all of them. The tracking information shows they should arrive this Friday.
I took the board to work and removed the failed rectifier.

I used the vacuum extractor for the first time and I think I put a little too much heat on the board. I did not fully lift a pad, but one of them is starting to come up.

I had my iron set for 660 degrees. I am confident I will do better removing the rest of the rectifiers. Here is the setup I used.

Following tggzzz's advice I decided to see if I can determine if something caused the rectifier to fail. The first thing I did was measure the AC from the transformer going to the rectifier. I measured it with my Extech 430 at 18.21 VAC.
I then decided to use my Leader LPS-152 power supply to supply the 20.6 VDC that the failed rectifier should output. First I disconnected the power supply from the rest of the oscilloscope and set my current limit to around 200 mA. I measured -14.977 VDC on the -15 VDC rail and my power supply showed less than 40 mA of current.
I then reconnected everything in the Oscilloscope and powered it on, again supplying 20.6 VDC from my bench supply. I measured -14.969 VDC on the -15 VDC rail and my power supply showed about 330 mA of current.
Best of all, the display came on!

I do not know of any other test I can do to make sure nothing else could be causing the rectifier to fail. My plan is to clean up the power supply board and replace all of the rectifiers when my parts get here on Friday then test the full functionality of the Oscilloscope this weekend.
Thank you.