I've done some research this morning with regards to parking. Unfortunately there is no simple answer.
I phoned the council - they no longer have any free car parks in Caernarfon and (conveniently) weren't aware of anywhere you could park for free, other than residential streets. They do offer a weekly fee, but this, as it turns out is more costly than paying daily.
I also phoned the Harbour Trust who own the large car park below the castle. They said that they don't have a weekly fee either, but that their daily fee is £4. You could to all intents and purposes park there for the week, explain your situation to the attendant and go back and forth each day to pay your daily fee. But any car left in the same spot in an open public car park for a whole week is going to attract unwanted attention.
I have been told that Caernarfon Football Club may allow you to park there for a week for a fee - I have yet to get in touch with them. I'll let you know as soon as I know anything at all with regards to this option.
I am planning to ask on social media as well. I have many contacts in the Caernarfon area and you never know, there may be someone who knows of somewhere secure which may be suitable. Knowing how many cars we have will be helpful. So if you could let me know if you are coming alone/on public transport/car sharing, then I can make my enquiry more specific. Diolch yn fawr iawn!