You can export a Bifrost scene to an Arnold ass file and then render it with kick.

Just use the -l flag (or the ARNOLD_PLUGIN_PATH environment variable) to point to the Arnold plugins that comes with the Bifrost install. For example, on Windows:

kick -v 5 -dp bifrost_aeroColors.ass 
-l "C:\Program Files\Autodesk\Bifrost\Maya2020\2.2.0.2\bifrost\arnold-6.2.0.0"

In the Arnold log, we see that Arnold loads the procedural nodes for Bifrost:

00:00:00 88MB | loading plugins from C:\Program Files\Autodesk\Bifrost\Maya2020\2.2.0.2\bifrost\arnold-6.2.0.0 …
00:00:00 100MB | arnold_bifrost.dll: bifrost_graph uses Arnold 6.2.0.0
00:00:00 100MB | arnold_bifrost.dll: bifrost_object uses Arnold 6.2.0.0
00:00:00 100MB | arnold_bifrost.dll: bifrost_multires_volume uses Arnold 6.2.0.0
00:00:00 100MB | arnold_bifrost.dll: bifrost_multires_implicit uses Arnold 6.2.0.0
00:00:00 100MB | arnold_bifrost.dll: bifrost_volume uses Arnold 6.2.0.0
00:00:00 100MB | arnold_bifrost.dll: bifrost_points uses Arnold 6.2.0.0
00:00:00 100MB | arnold_bifrost.dll: bifrost_implicit uses Arnold 6.2.0.0
00:00:00 100MB | arnold_bifrost.dll: bifrost_polymesh uses Arnold 6.2.0.0
00:00:00 100MB | arnold_bifrost.dll: bifrost_blocks uses Arnold 6.2.0.0
00:00:00 100MB | loaded 9 plugins from 1 lib(s) in 0:00.06

In other applications, like CINEMA 4D or Houdini or Katana, you can do the same thing by setting the Plugin Search Path.