This project saw some success today! I will say it was the most anxious I have ever been soldering anything. The cost of failure was rather extreme by my standards.
In the end, I elected to design and machine two special fixtures that would hold the PCB, provide thermal protection for the existing connectors, and provide a way to apply heat safely. Certainly the first time I have ever machined parts to solder a single connector. Lots of consideration and half a day at the CNC - it was game time.
I put all the parts in the convection batch oven and pre-heated at 100C for about an hour followed by 150C for 5min and 200C for 3min. At that point, I transferred the fixture to the bench where I had a special holder ready and 2 hot-air systems. One hot air was setup to heat from the bottom of the PCB, the other was handheld and focused on the work area. All was being watched with a microscope so I could see the moment the solder melted. Lots and lots of tacky flux was pumped in and I kept some motion on the hot air nozzle to avoid melting the connector.
Initially, I was planning to remove the connector, clean the pad/pins, stencil paste, and re-flow. Once the original solder flowed, I decided not to remove the connector and see how it looked. With all the flux, the solder wicked nicely on the connector pins leaving a nice fillet and a shiny finish. At that point, I decided not to remove it and examine the result more closely. After a clean-up and examination under the microscope - all pins flowed and looked great.
The fixture protected all the other connectors well and the end result looks like a brand new board. What a challenge. It was fun.