The reason that many popularizations describe "mass" as increasing with velocity is based on special relativity, and has nothing to do with gravity. I suppose I should look up what's been done in the way of formal experiments more carefully, bu at this point t instead I'll just remark that CERN, for instance, simply wouldn't work if the relativistic effects on "mass" weren't accounted for properly.
As far as your question goes, it's wrong to think of applying Newton's gravitational law with a modified expression for "mass" to get a relativistic version of gravity. To explain the right approach takes a book. The good news is that there are explanations available at the advanced undergraduate level, the topic used to be accessible only to graduate students. The bad news is it turns out that the simple model of gravity as a "force" is insufficient to explain the effects of relativistic gravity, one needs to get into the geometrical aspects of space-time. For instance, time dilation due to gravity is well known and experimentlly verified, but does not follow from any model of "gravity" as a force.