I ended up just redefining everything like so:

\chapter*{Appendix}\label{chap:Appendix}
\addcontentsline{toc}{chapter}{Appendix}
\renewcommand{\thesection}{A\arabic{section}}  
\renewcommand{\thetable}{A.\arabic{table}}  
\renewcommand{\thefigure}{A.\arabic{figure}} 
\pagestyle{plain}

..which gives the (mostly) intended result.