Since you've already told us that the 'L' button is the most import button for this game. It sounds as though it may be worn out from repeated use. It happens very frequently and the button can easily lose the "click" feature. This link is of a DS Lite Shoulder button I broke down to inspect but it has primarily the same features in your 3DS XL. The second item is a little convex metal disk that "clicks" when you press down on it and springs back up. When it clicks, it makes contact with all the metal pins in the housing and sends the proper signals to the game. When it wears out, it can either be dented or flattened to far.
The jamming could be a combination of not springing back up properly and either a couple different possibilities: the spring is rusty, there is too much dirt/gunk/debre holding it down, a dent could cause the plastic to hold it in place too. There are a multitude of various reasons why it's getting stuck.
To fix all of this, your best bet is to order a new 'L' shoulder flex ribbon off eBay and get a small phillips head (#000 size) screwdriver to open it up. Once open, clean your shoulder button area with rubbing alcohol and replace the flex ribbon. You can follow this guide to accomplish it: Nintendo 3DS XL IR Board Replacement to step 8 then just remove the plastic and such to get at the ribbon. Be careful not to lose the pivot pin or spring when removing those items.
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