Have just written a blog post about this: https://forge.autodesk.com/blog/use-custom-fonts-design-automation
The bottom line is that you can use System.Drawing.Text.PrivateFontCollection in order to load your custom font before opening the drawing and that should make things work:
public void RunWithArguments(Document doc, NameValueMap map)
{
LogTrace("RunWithArguments called with v7");
string bundlePath = map.Item["_1"] as string;
string contents = System.IO.Path.Combine(bundlePath, @"DaSamplePlugin.bundle\Contents");
string dwgPath = IO.Path.Combine(contents, "AssetLibraryTest", "CustomFontTest.dwg");
string fontPath = IO.Path.Combine(contents, "AssetLibraryTest", "DancingScript-Regular.ttf");
PrivateFontCollection pfc = new PrivateFontCollection();
pfc.AddFontFile(fontPath);
doc = inventorApplication.Documents.Open(dwgPath, false);
doc.SaveAs("result.pdf", true);
}