A few suggestions:
You need a 3/8" plug cutter
Get a tapered plug cutter. Then just tap them in with a hammer and they'll wedge themselves, no need for any adhesive. I've applied dozens this way and never had one come loose. Never.
Use a chisel to trim the plug nearly flush to the handrail, then sand. Look at the grain before paring the plug, go with the grain to avoid having the plug break off in the plug hole.
I was thinking of getting round dowles
I've never seen a teak dowel (or mahogany either). Regardless, if a dowel is used, when inserted in the hole, it will be the wrong direction and you will expose end grain to the weather. A plug should match the grain of the wood into which it is inserted. Dowels are rarely the dimension advertised and often somewhat smaller. Using a dowel would result is a really loose fitting plug. I don't use dowels for most woodworking applications, waste of time and effort.
dont you need a certian debth to have the bung stay.
If a tapered plug is used, the min depth of the hole (with screw inserted) is the depth of the plug. I can't recommend tapered plug cutters enough. I threw all my straight plug cutters away years ago.