According to the Strata Community Australia (NSW)’s “Who’s responsible ..?” fact sheet. “Any pipe, cable, ducting, plug or the like that is for the exclusive use of the lot [and is] outside the lot or in a wall, floor or ceiling that abuts to either common property or another lot is the owners corporations responsibility.”
That means, apart from anything else, you have to allow access to it. However, charging $5000 to shift the cables sounds excessive to me. Get your own quotes but be prepared to pay half (or whatever your share of unit entitlements is). At least then you know your momentoes will be safe and your private property will indeed be private.