That first picture, I was thinking I don't think a normal camera would even see that spot with visible light.
The camera compensates for bright 'light' by adjusting it's exposure. However, it does this over it's whole image so the bright spots become normally exposed while the darker spots become underexposed.
That branch reflects a huge amount of light, so even with that gone the reflection off the side of the house will still make it darker than you'd like. You need to experiment with this. If you do move the camera just keep in mind the exposure thing.
You could install an
IR illuminator to help if you don't want to move the camera.
Smart IR in my experience doesn't do anything or not noticeable enough to make a huge difference.
The BLC is handy. However, you can't keep the camera in auto-switch mode as the BLC stuffs up exposure during the day if you set it up for night conditions. (I thought you could set them independently with Auto-switch, however I can't seem to on my camera).
You would need to set it to switch manually at a specified time.