Hey @Jeremy_Scott!
You're correct in that the best way to handle indexing errors is in the logic making the indexing requests. It's going to be cumbersome to catch them by inspecting data after the fact.
There's no way to filter in the API Logs UI on indexing requests where one or more documents had an error. The only way I can think of is manually searching over the Elasticsearch index that contains the API Logs data. Note that based on your App Search deploy's ILM policies, I'd expect the data to be routinely aged out. To complicate matters more, the index name will change often based on that same ILM activity, so you'll probably have to search over a pattern instead. Finally, because we're referencing "under the hood" access to data, there's no guarantee that this will continue to work in new versions.
On a fresh Cloud deploy, this is where I could find document indexing errors from my indexing API requests: .ent-search-api-ecs-ilm-logs-production-2021.09.06-000001.
I hope that helps!
Ross