I am not sure if you ever found an answer but I did work on this myself for a few weeks and found a way that I thought I would share in case you or anyone else are facing this problem.
There is two parts:
You'll need to edit the AssetPortalBrowser.aspx to allow anonymous access.
<asp:Content contentplaceholderid="PlaceHolderDialogBodyMainSection" runat="server">...
protected override bool AllowAnonymousAccess
Deactivate Limited-access user permission lockdown mode
Now once you browse to the page with the asset picker you the sigin pop up won't come up when clicking on the browse. However you will still get the 403 if you just directly browse to the http://< site name >/_layouts/15/AssetPortalBrowser.aspx as that is missing query strings.
Note: With Limited-access user permission lockdown mode deactivated, all system pages that allows anonymous access will now be available i.e. http://< site name >/_layouts/15/viewlsts.aspx.
I used an http handler to redirect a user if they were into the viewlsts page via the public website url. If anyone knows of a better solution than this I am up for suggestions. But from my trials/research you need Limited-access user permission lockdown mode deactivated to access the document library view.