Nintendo (technically) won't give you any warranty coverage (which with Nintendo, usually means you get a free repair/replacement) if you don't have your receipt, because they can't know if the warranty period still applies to you if you can't prove when you bought the product. That's just common sense. You'll find that policy pretty much anywhere you go.
What people are saying is that you can purchase a second Wavebird, switch the two, and say it was the newer one which broke. It's slightly "dishonest", but you do say that you bought the broken Wavebird somewhat recently, so very few people would argue that it's "wrong". It does mean that you'd have to pay for another Wavebird, but you will get a second Wavebird out of the deal, and you can just chalk it up to the price of losing your receipt.
I don't know if you can fix the Wavebird by yourself using parts from a regular controller. I think the rumble motors (which the Wavebird doesn't have) are tied into the analogs on the regular controller. You'll probably just end up breaking two controllers (your Wavebird and the regular one) if you try. And if you take out the tamper-proof screws, you may NEVER be able to pull off the swap trick by buying a new one (it's why they make the screws "tamper proof").
And a regular controller isn't much cheaper than a Wavebird anyways. No sense just "using it for parts".