Our Google Ads API integration supports sending Customer Match parameters such as emails and phones to the API. Assuming you're capturing them for the Shopify customers, we'd be able to receive webhooks from Shopify when an offline sale occurs and send this conversion to Google Ads API, allowing Google Ads to use modelled conversions and attribute these purchases to its users by phone and email.
If you have a SquareSpace website with a lead or a sign-up form, this would work even better: we'd be able to attribute purchases to the specific ad click & visitor who completed the form using the same email/phone you have in the Shopify POS. In this case the sales could be attributed to ads with much better precision.
The implementation details would depend on how exactly the customers purchase.
To set up a trial account, please contact as, sending a link to your marketing website, as well as whether there are any special requirements with regards to how Shopify sales should be treated (how products are setup in Shopify, what conversion events you're interested in, how much time there's between an ad interaction and the sale typically etc).