Don't let the rules get in the way of the fun. Especially if you are the DM / campaign creator.
D&D was never meant to be a strict game where the rules are king. Leave that for competitive tabletop games like the war games it was born from.
If you need a character like that, let them. Should anyone call foul, kindly show them the golden rule.
Also, character's backstories aren't set in stone. It's what the character believes happened, but can we be truly sure? Did he naturally lose his eye and hand (or was born with), or perhaps some Wizard that worships Vecna set up the whole thing with his illusions and charms.
Think about that. For a character to obtain two artifacts out of a set, some fate shenanigans must be at work. I once heard an entire trilogy of books around a single artifact became very popular and even made it to the big screen as a trilogy of movies, fancy that!
I understand the OCD of having the story be truthful to the intentions of the designer to a tee. To do less is to compromise the vision.
I also know very well that I have one thing in my head, and then it becomes another entirely different thing on the computer screen / paper because I need to use those bony appendages to bash plastic cubes (or scratch carbon rods on plant fiber) to transmit them. Talk about a lossy data transfer protocol. (I really wish I could just plug an USB cable to my temple but I digress).
By the rules, if the facts stated are all true and immutable, you cannot have your character with the evil McGuffins. Don't let them win over the fun and a good story / campaign.
matszwecja has it right. There's magic that allows one to regrow an eye and a hand, only to have them excised for the sake of power.
Go ahead and make your campaign. Keep what the characters believe and what really happened separate. Life is full of inconsistencies due to unforeseen / unknown factors. Stories should mimic life and stir feelings, not be bastions of logic. If they can be both, great. If not, well. Shrug and move on.