Unfortunately, there’s no direct method to pass weight maps between the two programs, combining them at different stages of your workflow—PixInsight for integration, APP for mosaic—should give you the best of both worlds.

Since my current problem is the transition between the two steps (separate integration creates the unequal weights, mosaic should ideally take them into account) that means I will need to rethink my workflow altogether. Thanks, though, you've given me food for thought!