On a technical note, the current taken by your lamp would be in the order of tenths of an amp (If that), so the flex you suggest would work. The 13A plug must have a 3A fuse in it.
On an ethical / legal note, if you are updating, hence modifying, the lamp, you are personally taking over responsibility for its safety. The lamp would have originally gone through a (hopefully) rigorous design and testing regime, and shown to be safe according to the standards of the time. By changing anything now, you invalidate that original verification. If it's your lamp, to be used by you, you might feel you are happy to take the risk of your own handiwork backfiring. But if its going to someone else, or be used by them, it's totally your responsibility to make sure what is done is safe (and prove it).
This really means technical competence and workmanship done to the proper standards, and then testing the updated item (eg PAT). Not knowing if "replacing the cable" involves screw terminals, or soldering, or modifying strain relief cos the flex is different, or if adding an earth will improve safety, it's impossible to advise.
Be sure to keep safe, and keep the lamp safe too!