While I agree with you, and I know other people's money shouldn't be spent towards you just to fix an accident, sometimes you have no other choice. I don't agree with apple having to pay for a customer mistake, but if someone's career depends on that person having their laptop, then it might be necessary just once. I'm not an immoral person, I just believe that sometimes there is no other choice except to be immoral.
I'm curious, what is the difference between what you propose and outright stealing something because you need it for a career? What you are proposing is certainly immoral and unethical and you seem to engage in a contorted degree of post hoc reasoning to justify your proposal while still maintaining the integrity of your suggestion. You are suggesting lying and engaging in fraud for an economic end, nothing more. Are you telling me if the OP's LIFE depended on it he couldn't scape together $40 for an external hard drive enclosure??
The OP has my deep sympathies. Laptops can be somewhat fragile things and we've ALL done stupid things that end up being very costly. I was tired leaving work one day, lapsed for a second and put a big scrape in the side of my car. The facts, however, are not in dispute: Through his sole negligence, he damaged his own laptop. There is no reasonable course of action except for him to pay for its repair or replacement. No one else shared any responsibility in creating the problem.
The Golden Rule is an excellent way to test the ethics of a proposed action: How would you like it if Apple lied to you? How would you like it if you had NOT damaged your laptop through spillage, sent it in, and they activated the LSI and said you had damaged it yourself? Of course the perceived injury to an individual is different than the financial cost to a company but that is about the consequence of an action, not it's ethical foundation.
Some people would additionally argue that lying hurts the liar. It makes the liar believe less in his or her own character. It makes the liar more suspicious of others. When you start acting in a way that is contrary to your beliefs, your beliefs have a funny way of changing. If the OP decides to do this and owns his own business one day, I wonder if he'll be a little more suspicious that other people (his customers, employees) are trying to cheat him because he's looked into his own heart and found it wanting.
If the ethical argument doesn't persuade, I doubt that replacing the LSI sticker is guaranteed to fool Apple. If you replace it with a white sticker, they may test and replace the stickers to see if they've been tampered with. They may have UV indicator ink. Who knows. They place the stickers there because they EXPECT some people to try to defraud them by denying exposure to moisture.
My suggestion would be to not make a poor situation worse. Get a HDD enclosure, get your paper out and defended, and repair your laptop when you've saved up a little bit of money. THAT's character.