No, the SS filler will not stick to the copper. You can use aluminum too. In fact, to get the effect you're looking for I would suggest a bigger aluminum bar (at least 1/2"). There won't be enough heat available to melt either.
If you're new at welding stainless, there's a couple things to know. The unprotected side of your weld will grow big black oxide crystals if left unchecked (ie, back purged). A close fitting chill bar (what you're talking about) will limit this as it limits the amount of oxygen available to the weld. If you want to eliminate the black crud entirely, you can back purge with argon. There's also Solar Flux which I guess you can stuff between your chill bar (copper) and the weld to take up the headspace but I haven't tried it myself. If you don't care what it looks like inside, but just want to prevent the weld from ending up intruding into the ID, your bar will work fine.
The second thing is that stainless warps, due to the two-punch high coeff of thermal expansion and (for a metal..) insulating properties. The shape is good for that but I'm worried they may end up a little bent. Take your time and go with short welds if you want to prevent warping. (Good contact between your chill bar and the stainless helps, but I don't think you can really have that with a piece of round.
Since you've got the SB, I suspect you've got some larger hunks of aluminum round around.
EDIT: jakeru you beat me to it!