Has anyone worked with SQLDelight and PostGIS together?

I don't have any experience with PostGIS and SQLDelight doesn't know about extensions

How would you approach handling custom types and functions through a custom dialect?

There are two ways:
Simpler: You can try extending the existing PostgreSql dialect with delegation e.g https://github.com/griffio/sqldelight-custom-dialect.
This maybe the quickest way to get some minimally useful functions working
Complex: Create an entirely new PostgreSql dialect where you can add custom language features in the grammar e.g https://github.com/Faire/sqldelight-cockroachdb-dialect

Is there any possibility of adding native support for PostGIS in SQLDelight?

I don't think that would happen in any time frame that would be useful to you - there are still many native built-in PostgreSql features e.g CREATE Trigger, CREATE Function, that are not implemented yet