I've tried many different combinations of inputting the code. With the hints from the tapestries with the words 'The Order' on them and watching the glowing gems, the sequence appears to be:
Goat (5) -> Lion (8) -> Eagle (3) -> Ox (4)
And the need to select the Ox multiple times (owing to the number of gems the Eagle is holding).
HOWEVER: These clues of order is an (unintentional) red herring.
I even considered how the L in Roman numbers meant 50; and how it might reflect in the puzzle in some way. But no, that was also a red herring.
The solution? Knowing the start and end animals and simply swap the order you visit the others.The developer seriously needs to correct the way in which they shine the gems!
Why is this annoying? Because this is a fabulous game let down by an unfortunate quirk.