How should I resolve my birding GAS?

To start with, I would ditch the 2X converter.  It appears to be seriously degrading your image quality and the reduction in light won't be doing you AF any favors either.  2X converters, in general, tend to seriously degrade image quality and you are often better off just cropping further from a 1.4X converter.

In terms of equivalence, your combo has a full frame equivalence of 176-800mm f/8-11.  That is quite long and quite dark, especially when paired with an older camera like the original M6.  Larry Rexley's recommended 1.5X teleconverter would put you at 132-600mm f/5.6-8, which is still quite a bit of reach.

I could be mistaken, but your post reads a bit like you are looking for a quick fix because you currently have some free time and a bit of free cash.  I would suggest taking a long term view of your gear and determine where you want to be a year from now instead of what might work this minute.  Maybe that means the R7 + RF 100-500mm, or maybe the M6 II + Sigma 150-600mm, or maybe you continue with what you have.

In the interim, I would recommend working on yourself.  Learn everything you can about the birds you are trying to capture and then use that information to put yourself in a better position to get closer to the birds.  Start with the easier birds that will let you get really close.  Even the lowly sparrow can make for a compelling image when photographed  from only a meter away.  Drop the teleconverters and see what you can capture with just the 55-250mm as this would be equivalent to what the 100-400mm would get you on your R.