I think a better alternative if you re set on this design is to use a 1 2 panel.
Attaching shaker drawer fronts.
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It will be level with the drawer after installed.
You could also run your drawer handle through the panel and use those screws to help hold the front on with some screws on the side into the stiles.
Then you d have more thickness to screw into.
Make sure the slight hole is only deep enough for the screws head.
Take the drawer and drill a slight hole on the 4 corners of the front inside of the drawer leave at least 5 1 space from the ends this is where the screws will be placed to add the front.
Hope this helps some.