I would suggest turning off the IS on your existing lens and trying it under a range of conditions. It's a cheap way to know whether the loss of IS will bother you much. I find myself shooting at 1/250th and slower fairly often when the light isn't great, so IS is a big deal to me. And no - my birds won't sit still for a tripod shot either.
I'm not saying this is the best choice for you, but my most common setup for birds is a 300 F4 L IS with a 1.4x II extender. That was my attempt to stay away from 5-digit lens prices. It's not too bad, but I find that I have to stop down to F8 to get acceptable* sharpness with the extender, and F11 is better. That means higher ISO and lower shutter speeds than I would like, most of the time.
*Okay, "acceptable" is relative. If the goal is to clearly identify the bird and keep a record of it, F5.6 is plenty sharp. If the goal is a print worth framing, it's not.
If I were shopping now, I would avoid anything that has to be stopped down below F5.6. The 400 F4 would fit the bill, but I'm not willing to give up IS.
I've attached an example of the compromises I end up making. It's a White-eared Jacamar on a cloudy day in the Amazon. You could argue with the exact settings, but here's what happened: ISO 3200, F8, 1/400 second, 420mm (300 x 1.4), on a 5DIII. Taken from a small boat, or I would have used a lower shutter speed and ISO.
It's a pretty noisy shot, but I didn't have the option to wait for a sunnier day. I could have bought the 600 F4, but then I couldn't have afforded the trip to the Amazon