I followed No_Sprk's advice (Thanks!!) and had a moment to investigate the Weber after work today.
At the start of the day it would idle barely and either backfire through the carb or die when I applied the throttle.
First I lengthened the throttle linkage so everything was more aligned. Then I followed the idle tuning notes from Weber very closely. After doing this I determined that the idle jet was small and causing it to run lean.
I pulled the primary idle jet and swapped it from a 55 to a 65. It ran a lot better so I swapped the secondary from a 50 to a 55. Then I took it around the block and it ran like an absolute champ.
I'm not sure if the SU HIFs are completely worn out or just poorly adjusted. Part of me wishes I spent more time trying to learn about them before I jumped to the Weber but for the deal I got on it I think I am going to be very happy with a brand new carb.
More goodies arrive tomorrow! It is so relaxing to come home from work and play around with something mechanical for a few hours.