As noted in comments, look into layering.

I did this in Photoshop, as I don't have Raw Therapee. You could do the same in Gimp etc.

Select the bird & separate it to a new layer, by copy/paste. Then you can treat each layer individually. Essentially you can push the sky darker & the bird lighter, without each interfering with the other.
I over-cooked these edits to show how far you can push it. It's all a bit crunchy because of the jpg start-point plus I didn't really take the required care to accurately select the bird. You could do a whole lot better from the original, though you may need to save it out as a 16-bit TIF to manipulate in this way, retaining the highest quality. I don't know of anything that can edit a RAW in this way.

Zoomed to bird, more careful selection technique & some feathering of the edges [no pun intended] could smooth this out nicely.

One additional trick. This type of edit where you are pushing adjacent areas hard in different directions tends to make sharp lines at the edges, worsened by the original bird in this case underneath. Use the healing brush to eliminate the bird from the lower layer, so you only have to concentrate on the upper for matching/blending.
This also allows you to use masks to in effect 'cross-fade' the edges rather than blurring them [I didn't do that here, as you can probably tell]

Incidentally, just for fun, this also allows you to then move the bird wherever you like ;)