Rick-
We all have different approaches to these problems. Here are my thoughts:
First, I think you can get through this with nice results.
Second, strictly from the standpoint of a drill wandering as it cuts through, I would recommend 1/4" over the 3/16ths.
My opinion about finding the center is this: use the smallest, shortest drill, with the highest spindle speed, no fluid. Stand looking down the rod after having measured, aligned, realigned, so you know it is in the center. Touch that drill to your axle, and if it is in the center, it won't walk to the left or right, leaving a polished trail behind. If the drill point walks off center, recenter the rod.
Using cutting fluid, reduce the spindle feed by one if you have a four speed and two if you have more.
You are ready to start cutting. Visualize the drill cutting through the stainless and concentrate on cutting, feeling continual movement as the drill cuts. When the drill is cutting, it is cooling itself to some degree. When it is not cutting, it is heating and destroying its ability to cut. The continuity of the feed/cutting speed is important. Once you feel it cutting, force it to maintain the same rate, until it fights you. Then pull the drill out, add more cutting fluid, and let it cool down if necessary. HSS drill should be fine, easily resharpened as necessary, but I would always start with a new one because if it isn't sharpened perfectly, it will walk and not come out 180 degrees from where it went in.
The next ghost waiting to bite you is at breakthrough/ Set your stop to make you stop and re-oil before you go for the final cut, and take that real easy from a feed standpoint.
I think you will get through this with good results, and the heavier than average drill press, is a major factor, gives rigidity.
Michael