So the correct path is "message.content.background_rgb"
Also, you can't send "rgb(4,107,109)" in JSON, because it's a string. Well, you can, but then you will have to parse it into three individual numbers 4, 107 and 109..
So you need to send three separate numbers - "background_r":4, "background_g":107, "background_b":109. And then use this code to set color:
JSON Set Path to "message.content"
Sprite set color to rgbex255(JSON.Get(".background_r"), JSON.Get(".background_g"), JSON.Get(".background_b"))
Another option is to pre-generate the rgb colors which Construct will understand and send them from the server. For example, if you run this small event
TextInput Set Text to rgbex255(4,107,109)
... you will see that it returns the number -4405244385279. So you can just send this number in JSON key "background_rgb" and then use this:
JSON Set Path to "message.content"
Sprite set color to JSON.Get(".background_rgb")