Thanks so much for your responses. I had already verified with Google Search Console, and have now set up Google Business. No result yet on the search bar, I'll also have a go at some of the other suggestions here and see if that improves things.
When trying to add my .co.nz domain it simply says that squarespace does not support that type of domain. I'm tempted to set up an entirely different site for my live wedding painting servies but I feel like that would be a too fiddly!