I used to have the Canon (7D2, 5D2, 1Dmk4) Canon 500mm f4 and the 1.4x converter worked on all of them fine. My issue with the 7D was that anything over ISO 800 seemed too noisy. I got frustrated with the noise levels and sold the camera. I do not mean to offend any 7D owners, this was my experience when I owned it.
Your sig lists that you currently have the D500. Very capable camera for birds, and noise is well contained and fairly easy to treat in post. I do not think you could go wrong with a 200-500, a 500mm f4 or, *gasp!* if you can find one, a 500mm PF. I find the 500mm PF can also handle the 1.4x converter, although you do lose some AF points.
Also, at least on the Nikon side, 500mm f4 G lenses seem to be at bargain prices. I recently sold mine so I could get the 500mm PF. After 2 months trying to sell the 500mm f4 I netted the exact amount needed to pay for the 500 PF. Look around, the 500mm f4 G is a fantastic lens. The 500mm f4 with 1.4x or 1.7x is an excellent birding kit.
Dave