The public IP address of assets is in most cases shared by multiple assets, and not "known" by the local asset itself. Lansweeper, for most asset types, will retrieve the information of all NICs of an asset and list them on the asset page of the asset in question, the IP through which the asset was last scanned will be listed at the top of the asset page and is the "main" IP address of the asset. Likely, the public IP address will not be among these IP addresses.
You could potentially enter the public IP address of an asset in one of the asset custom fields that are available. Asset custom fields need to be manually filled and can be set up under Configuration\Asset Pages.
In cases where multiple assets share a public IP address, for instance in the following layout: Asset -> Switch -> Firewall, you could potentially just enter the public IP address in a custom field on the firewall. The public IP address could then be retrieved for a specific asset by starting at the asset page of the asset, clicking the switch it's connected to and then the firewall the switch is connected to.
To create a master list of Public IP addresses you'd have to make sure you can generate an asset for these IP addresses by adding them under Scanning\Scanning Targets and then group them together with related assets via one of the asset grouping methods, likely a static asset group. More information on grouping assets can be found
here.