OP properly attempted to make his claim to AA upon arrival at LAX and was improperly referred elsewhere. AA's failure to accept the claim on the spot ends the timing issue.

The Montreal Convention protects AA, not OP. It sets the maximum liability, not the amount OP will recover. OP will recover the fair market value of the bag and its contents up to SDR 1131. Fair market value may be a lot less than replacement cost, all depending on what OP lost and its condition. By way of example, a 2-year old $100 shirt isn't worth close to $100.

Far better to make a claim on your insurance as that will generally pay up the full loss and quickly. Insurance will deal with AA and you won't be bothered.