Apply in Photoshop Image > Image size > 150 PPI, NO resample. Then save as PNG. Be sure you have color mode = Bitmap.
Here's my example in Photoshop. I started a new image 300 x 300 pixels, color mode = bitmap with bit depth=1.
I decided with no reason except you have mentioned number 150 let it have 150PPI resolution i.e. the print size I selected was 2 in x 2 in. That selection is arbitary and doesn't affect at all the image content.
As well I could select say 600 PPI. In that case someone who sees that PPI=600 and the size is 300x300 pixels may think that I perhaps want to print it only as 0,5 inch wide and high and continues: Proves nothing, PPI can be changed if needed and that does not affect what the image contains.
Here the PPI is set to 150 in Photoshop:
FastStone image viewer (freeware) digs from the metadata resolution 150 PPI. It says 150 DPI instead of 150 PPI which can make some people angry, because DPI is also printing term and means there a different thing.
You see the button DPI. By clicking it I can change the number to 600:
Reopening the image in Photoshop and checking the image size gives now height and width = 0,5 inch:
The image has stayed all the time same: 300 x 300 px PNG, color bit depth =1.