I've held up to 8" long material (1 1/2" dia, I believe) in the chuck of my mini-lathe and drilled in the end, but 15" long as way too long for the mini lathe since it won't pass through the spindle. If you have no other machine tools at your disposal, then clamping it to the carriage is probably your best bet.
I once made a simple milling fixture once for the mini-lathe that was a flat piece of 1/2" steel. It had 3 holes into it to mount where the compound was - 1 through hole to line up with the center stud in the compound clamp and 2 through holes for threaded bolts to attach to the clamp. Then I drilled and tapped holes where I needed them for clamping.
Anyways, once you have a plate like that set up on top of the cross slide, you could attach your 1" x 15" shaft to it with whatever clamps, v-blocks, vise, etc. you have. Use the cross-slide to line up with the spindle. (You'll need to shim the stock shaft up when you clamp it, since you don't have an up/down adjustment.) Then do as you were thinking - mount the drill bits to the spindle using whatever lathe chuck, drill chuck, collet chuck, etc. you want. Use the carriage to move the shaft towards the drill bits.
Andrew