You are correct, my test was incomplete, my apologies.
I've tried this again and added more combinations, which would get closer to your configuration I believe. It appears the symbology I've set in ArcGIS Pro still aligns with what I see in the Template Picker.
Before going down the customization route, I would suggest trying a few things:
1. Add the Edit widget to your application and see how the templates display there. They should display the same as in the Smart Editor. If they don't, please contact Esri Support so they can help troubleshoot the problem with Smart Editor.
2. If the templates are the same in the Edit widget and Smart Editor, republish the service and create a new web map and application, and make sure the service symbology is not updated after being published. Any changes you make to the symbology after having published the service may affect the way this is displayed in the template picker. If you did make changes in the web map, you would need to make sure those changes are saved against the hosted layer / service endpoint - but the behavior there could be inconsistent.
3. If still no luck, please contact Esri Support, as it is possible you are experiencing an issue for your specific use case, or there could be a known bug with an available workaround.
Regarding modifying the code, someone would need to customize the template picker which actually comes directly from the JSAPI and is a core component rather than specific to the Smart Editor. If you want to modify it, you would need to grab a local copy of the API. It is not the most straightforward process. These links may be helpful:
Using Bower for Custom Builds of the ArcGIS API for JavaScript | Guide | ArcGIS API for JavaScript 3...
TemplatePicker (legacy) | API Reference | ArcGIS API for JavaScript 3.38
custom template picker to have only specific item... - Esri Community