15m shouldn't see a lot of voltage drop. I used 6ga 3C wire on a longer run when I did my garage, feeds 240v to a panel fused at 60A.
If you have metal studs, you need to have a grommet at each point the wire passes through them. If you're using 2x4, you'll probably need a protector plate on each stud since there is a minimum distance back from the front the hole has to be.
Assuming that you have a homeowner's permit to do your basement already, adding the cable would be covered under that. But, you would not be allowed to have an electrical contractor do the work for you. There really is no need for one though, the termination in the panel and at the plug in the future bathroom are very easy to do.
See Crank. See Crank Walk. Walk Crank Walk.