I've been a
builder for 39 years so I'll present my opinion. It is extremely difficult to countersink and then plug
wood that is only 3/8" thick. By the time you are deep enough to set a plug the screw is almost all the way through the bottom of the wood. Extremely thin plugs don't hold that well because they are mostly gripping the metal screw instead of the wood counterbore which is what they are designed to do. How about trim
head finish screws instead. Predrill just slightly over size of the shank but smaller than the
head and then putty. You could whittle a very tiny plug for these as well which is pretty easy. Option B would be to just glue and clamp. Contruction
adhesive works good for this. Be especially carefull about glue squeeze out because it will fill the grain of the wood and will be extremely hard to remove if allowed to dry. A wet rag while the glue is still wet works well. Bob