Yes, when using a custom template you can just supply that as the base to auto_warp.py (or other AFNI programs). Do check to make sure that you've specified a SPACE and the correct VIEW (tlrc) on your custom template first so those carry over.
There's some concerns with warping stats files, since interpolating statistics could be problematic. The current AFNI recommendation is to wrap during the processing, before stats are created. That said, plenty of other packages do exactly this.