I have a Sigma 150-600 C lens but it's not going to get me the kind of really diffused background as say a Canon 500mm f/4. That's why I want to know the best option on a used budget around that price bearing in mind as much diffused background as possible and as much reach whether it be with TCs or not.
Your choices are a pretty much limited to
second hand Canon300mm f2.8 plus converters, Canon400mmf2.8 plus converter, Canon500mmf4, Canon600mmf4. New you could consider Sigma 120-300mmSport ( I had one and it was excellent bare and with a 1,4x TC but slow with a 2xTC) or the new Sigma 500mmf4 at £5000 (check out reviews). All of which are excellent and all weigh at least 2-3 time that of the Sigma 150-600C you currently have so a monopod or tripod with gimbal then starts to make sense. If going for the shorter focal lengths are you happy swapping TC's when needed in the field.
I had a Sigma 150-600C then bought a Canon 500mmf4 and loved it so sold the Sigma but after a few months I started to realise the 500mmf4 has drawbacks which are size and weight, so I then spent a fair bit on a decent tripod and a decent gimbal head which then made it even less useful as a pick up and go lens so I ended up getting another 150-600C which I now use as my main lens. The 500mm comes out when I know where I will be walking and it is not miles and I want that extra bit of IQ or need f4. I do hand hold it a fair bit for birds in flight but notice it the next day in my lower back. It is an excellent trackside lens on a monopod for bikes but a bit long for cars on a 1.3x crop body well at least at Brands, I have not done an airshow with it but would probably take the Sigma if I had to choose.
I don't think I would go out and get a 500mf4 or 600mmf4 just for smooth bokeh as the Sigma does a decent job if you get the background right in the first place. Where the Canon 500mmf4 wins is f4 in low light, I do a fair bit of that with owls, the image quality is superior if you nail the focus, focus is a lot quicker, brighter view finder and it looks the nuts. Downside over the Sigma is 1.9kg extra weight, dated OS which is ok but the Sigma's is a lot better, missing 150-499mm and 501-600mm focal length, the need for a monopod and or tripod with gimbal to really get the most out of a long photography session, minimal focus distance of 4.5m over the Sigma 2.7m, you need a much larger bag, the cost and trying to hide it from the Mrs.
This is one of your shots off flickr
https://www.flickr.com/photos/130378054@N06/35329782596/in/dateposted/ at 300mm f5.6 with your sigma and there is nothing wrong with the bokeh imo or with the subject having that pop out look.
I also picked these out, same place different lens but shows if background is far enough back and subject is close aperture f4 or f7.1 still works.
Canon 500mm@f4, 1/1250 ISO 400 on 7Dii
Barn Owl (Tyto alba) by
Martin Billard, on Flickr
Sigma 150-600mm at 600mm @ f7.1, 1/1000, ISO 2000 on 7Dii
Barn Owl by
Martin Billard, on Flickr
Regardless of what is said if you have £4000 and fancy a 500mm or 600mm prime just get one, if you regret it, just sell it I doubt you will lose on it.