1. is far better.
Student or Learner
I'd like to ask a question about word order.
Between the two setences below, which do you think sounds natural?
1. The image we have of the Royal Family is gradually changing.
2. The image of the Royal Family we have is gradually changing.
In your example, both sentences mean about the same thing. In another situation, they may be very different:
'A photo I took of the car...' You took a photo.
'A photo of the car I took...' You took a car.
I think the original also has that problem. Is the topic the image we have, or the Royal Family we have?
***** NOT A TEACHER *****
Hello, Optimistic Pessimist:
If you will (mentally) replace the deleted relative pronoun "that" and then read the sentences ALOUD several times, it may help you to decide which sentence sounds smoother to your ears. (I will keep my opinion to myself.)
1. The image + that we have of the Royal Family + is gradually changing.
2. The image of the Royal family + that we have + is gradually changing.