Gluing a C++ implementation of ANTLR into GCC so that GCC will call it is likely to be the easy step.
[Don't expect to be easy; GCC wants to be GCC, not your pet. You might get some help from GCC Melt, a package for interfacing to GCC machinery.]
The AST produced for an arbitrary (e.g., your custom DSL) language doesn't "just move (easily)" to a C AST or to the GCC Gimple (not GIMBLE) framework.
You will have to build, in essence, your DSL-AST to C-AST translator, or your DSL-AST to Gimple translator. There is no a priori reason to believe that building such a translator is easy; for example, you didn't tell us your DSL was "just like C except ...". So, you're going to have to build a translator. In the absence of evidence this is easy, you'll have to translate your DSL concepts to C concepts. The better ("non C-like") your DSL is, the harder this is going to be.
This SO link discusses the issues behind translation in more detail: What kinds of patterns could I enforce on the code to make it easier to translate to another programming language?