If you can take a hit on the weight of the PNG files you may want to simply use "save as" instead of "save for web".
This will save the file in the same folder as the tiff (Make sure you activate that preference in PS under File Handling)
Then in Lightroom all you need to do is synchronize the folder.
Also from Lightroom delete the unnecessary tiff files.
Note: In you action you may want to include a resize image step.