Snowbro, there is no way to paint a sintered base. The patterns you see in bases are either laid into the bottom lamination and covered with clear base, or colored base material is CNC cut and inlaid with a pattern. You can have your pattern custom manurfactured in a new pair of skis, but even then, most of that is done in topsheets. Here is an article that might help you. LINK
Are you sure it's paint? When I was a kid I had skis like that with bare wooden bottoms. We would buy a little can of ebonite (what bowling balls are made of) that came with a brush built into the cap and paint it on the bottoms of the skis. It would leave a fairly smooth and very hard black finish that we could then wax.