Lighthouse flags pages that don't have a custom splash screen:
In the Lighthouse report UI the full PWA badge is given when you pass all of the audits in all of the PWA subcategories (Fast and reliable, Installable, and PWA optimized).
Chrome for Android automatically shows your custom splash screen as long as you meet the following requirements in your web app manifest:
nameproperty is set to the name of your PWA.
background_colorproperty is set to a valid CSS color value.
iconsarray specifies an icon that is at least 512x512 px.
See Adding a Splash Screen for Installed Web Apps in Chrome 47 for more information.
While Lighthouse's audit will pass when a single 512x512 px icon is present, there is some disagreement about what icons a PWA should include. See Audit: icon size coverage for a discussion about the pros and cons of different approaches.
Source code for Is not configured for a custom splash screen audit