(1) Beginning late 2001, BMW installed a so-called "smart" Starter Relay on all K1200LT - this newer type Starter Relay will not execute (trigger) the starter if the battery voltage is below a certain threshold. BMW has not published the exact low treshold value but experience shows this to be roughly between 11.5 and 11.8 volts.
IT IS IMPORTANT to test you battery voltage with KEY-ON, before you hit the starter - the value "at rest" with KEY-OFF is pretty much meaningless as this is not HOW the Starter Relay internal circuitry is measuring it.
SO.... open the seat, get ready with your reliable multimeter, turn Key-On, wait 5 seconds (typical scenario in real life so as to let the ABS modultator self-check AND the fuel pump prime) then take the voltage reading. If the battery is below the threshold, the Starter Relay will not execute , not even a faint click - neither will you see any warning on dash. I would charge it overnignt AND get the battery load tested.
(2) If the Battery reading is above 12 volts in conditions mentionned above, then the next troubleshooting step will be to check if you can see the RED engine temp warning after Key-On (before you hit starter). This warning is related to activation of the safety starter interlocks (kill switch, neutral, side-stand...). On a 2003 like you have mentionned the engine temp warning is as shown below...
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