Except that it does gravitate more. Avoiding the confusion that it doesn't is one of the conceptual benefits of relativistic mass. Gravitation is dependent on the stress-energy tensor, which contains... drum roll... relativistic mass, not rest mass!
For that matter, you've already noticed that the Lorentz transform becomes nonsensical as v → c when starting from rest mass, whereas the transform works just fine when using relativistic mass as the starting point, and nothing blows up, and you can avoid any confusion.
