I have found what was causing the issue...
I was testing the various Compatibility Options and saw that there were some extra files in the Fallout 4 game folder/directory.
Looking at the files, they were related to ReShade, a utility that is similar to SweetFX, where an injector creates visual effects/changes (such as Contrast Enhancement, Film Grain, etc) for games. I removed these files (so that the game folder is as unchanged as the Retail game would be) and then the game started without error, with Bandicam running in the background (the issue was not occurring)!
While this solves what was going on, please accept my apologies that I had forgotten I was doing some testing with that visual effects utility/addon last year, and those files were still there. It was so long I completely forgot about it and that it was present. Sorry about that...
However, perhaps this may help those that run into this same issue - if the game is 'crashing immediately upon startup, when Bandicam is running already' - it is possibly due to a utility or addon that is utilizing Direct3D to create or simulate visual effects.
I thank you for your sincere and speedy Support in helping to find the cause of this 'odd issue'. Of course it wasn't Bandicam! haha
(just kidding - but thank you for your assistance.. marking as "Solved"). Keep up the great work!
Actually, since other Game Recording programs did not have an issue with this Direct3D effects injector utility (the game started with them running in the background, but did not with Bandicam running in the background), perhaps it could still be mentioned to whomever may be concerned at Bandisoft...
(Some files it used were dxgi.dll, and corresponding .ini files)
If you wish me to perform any tests with that utility (ReShade/SweetFX) and Bandicam, if it helps you at all, I would be happy to do so.