Birds are highly visual creatures. In general, they don't have a "critical distance" closer than which you cannot come, they have a "critical visibility factor". You can get much closer to a bird, especially a small bird, in bad light than you can when everything is bright and clear. Now you can think of this as one of life's cruelties, but it's more useful to see it as opportunity. When the light is poor, with care, you can get very close indeed to your subjects (or rather, you can allow them to approach very close to you). USE THIS! When the light is bad, get closer, and you don't have to crop as much. You can use a higher ISO AND get a cleaner overall result even so.

Now I'm a big fan of fast shutter speeds for bird work, but I'll let that speed drop as far as I dare when the light is bad - typically, I draw the line at around 1/250th with the 600/4. I can go lower than that handholding the 100-400 II - it's an amazing lens with brilliant IS - but don't usually do that birding.

Remember, you can use higher ISOs if you get close enough to avoid cropping. Close is good.