Yes
Mage hand (Basic Rules (2014)) says:
You can use the hand to manipulate an object
Surely, that includes using a scalpel, at least assuming the patient is unable to resist.
Surely a mage hand could wield a knife to cut a rope or a potato, an incapacitated patient is not much different.
You say "it's been said that mage hand can't be used to hurt other creatures". There's no support for that in the rules. As Kirt points out, that is not correct, it just can't attack.
But surely it isn't better at surgery than you are, and likely worse
There's nothing about mage hand that would make it better at surgery than you are, and since you are controlling it, but indirectly, it seems the DM might likely say that it is actually worse at it.
But what are you really trying to accomplish here anyway?
You say:
. . . we have now been given the task to make her pretty. The DM explains to us that this princess is quite hideous, with very bad hygiene and boils that would make a hag look pretty.
Bad hygiene is bad hygiene. Plastic surgery won't fix that.
Boils are a medical condition. Perhaps you can consult someone skilled in medicine? Surely that makes more sense than home surgery. Or maybe consult a local healer, who may be able to concoct a salve of some sort.
Furthermore, you might consider arguing that the boils are an infection, which is a disease, which should be curable by magic. That won't make the princess "pretty" by some arbitrary criteria, but at least it would cure the actual disease. Although that brings us back to hygiene.
Has the princess met the dragon?
You say:
who wants to marry a dragon
You don't mention if the dragon is amenable.
The dragon's opinion might be important
It seems unlikely that you are going to be able to coerce a dragon into marrying against its will, and if you can, beauty won't matter.
The dragon may be willing to marry the princess regardless of her looks. Maybe the difference between a pretty goblin and an ugly goblin isn't that significant to a dragon. I mean, the dragon's still a dragon, and the bride is still something besides a dragon.
There are all sorts of reasons the dragon might be willing to take the princess as-is: dowry, political advantage, inheritance, a snack later, who knows. For one thing, being a princess implies royal connections, maybe even a kingdom, or at least large tracts of land. Maybe the dragon keeps humanoid spouses as decorations, and has dozens back in its lair. Nothing says the dragon need be monogamous.
Or . . . maybe the dragon and the princess could . . . I don't know, get to know it each other. After all, they say there's someone for everyone.
In any case, it seems the dragon's opinion here might be paramount. After all, beauty is only skin deep, but being a dragon goes to the bone.