Well, from your thread title, you seem to be assuming that the problem with your shoulder button is in the spring somehow, but from my experience and based on your description of the problem, it sounds more like the microswitch for the R-button has become dismounted from the PCB. That's a somewhat common problem with DS Lites. Unless you're very very handy with a soldering iron, you won't be able to fix this problem. So the solution would be to swap out the entire lower half of the PCB from the black Lite into your white one. That should be possible and honestly not that hard once you've got the system open, but since the black Lite also has a problem with touchscreen, you'll want to also keep the working touchscreen from the white one and connect it to the PCB from the black one, which may be harder (I don't know, actually, since I've never even tried to detach the touchscreen from the DS Lite PCB).
EDIT: Actually, Tanas may be right. Maybe you just have some lint in the microswitch mechanism and then WD-40 (a very tiny bit, I imagine) or compressed air would do the trick. That's the most common problem with DS shoulder buttons. I guess I'm just pre-occupied with the part becoming dismounted since that's what happened to my DS Lite.
In my opinion, the best solution is just to open up the system and see what exactly is wrong with the shoulder button. Maybe it's not detached from the PCB afterall and the fix really is simple, but if it isn't and you don't want to deal with removing the touchscreen, then just close it back up and send it in to Nintendo.
Yes, technically, opening up the system voids your warranty, but I don't think Nintendo checks or has any way of checking, so as long as you don't do anything dumb like leave a wad of chewing gum inside the system or something (actually, that probably wouldn't even matter either), you should be fine.
Same with the issue of a FlashMe'd DS. FlashMe may or may not void your warranty (I don't think it's clear from the warranty description), but either way I've never heard of anyone having their warranty repair request turned down because of a FlashMe installation and lots of people have sent Flashme'd DSes in to Nintendo. Besides, no one even uses FlashMe anymore, do they?
...word is bondage...