if you are constrained with some framework and these are the only two options then
Leaf's and Doddy's answers are the best solutions to your dilemma.
However if you are not constrained and you can really design to optimize user experience I would suggest the following states:
For editing the inputs must be slightly 3D, to give some depth, and the user will know that they are editable
Having the data belong to the same model and also have such a distinct look and feel, either be in a textbox or either look like a label is not consistent and pleasing to the eye.
One solution is that they non-editable fields have a different color and a contrasting background color and these are completely flat. Flat is like printed out, end of story, nothing more to be done here.
For desktop with a cursor situations:
Don't forget to set the cursor to a hand cursor. The action is that when the user clicks on this flat field you get a copy of the text in the clipboard. Also make sure that there is tooltip that informs the user of the copying-to-clipboard action in order to avoid losing already existing clipboard data.