I have then paid too much of attention to the wording containing "shall",
Yes.
Is there any right contact where I can submit my idea to? Perhaps the office of Mr. Picardo or the straight to the treasury?
I do not think so. Mr Picardo will have little to do with coins. The Treasury or people like Joe Bossano will have more responsibilities in regard of coins. However, since it is extremely difficult to get hold of anyone at the government of Gibraltar as, from my personal experience, the Treasury and the office of Mr Bossano virtually never reply to emails, I am really not sure if it would be worth the hassle. But anyway, feel free to give it a try but know that the chances they will bother to read it, let alone do something with your idea, are slim to none. It is a good idea though and I think a coin like that would fit well with the other coin and would prove popular amongst collectors.
Alternatively you can maybe send your idea to mints like Tower Mint or Pobjoy Mint. Of course you are then unlikely to get credit for it would they indeed issue a coin like that but I think the chances of success that a coin like the one you described will be issued is then much higher. In general the mints have better direct contact with the Government.