Centering captions in document that contains A3 and A4 paper sizes

There is more to a page layout, then just the well known parameters like \textwidth. For example, internally TeX uses \hsize for building paragraphs. The best is, to leave this to the geometry package.

Unfortunately, \newgeometry doesn't allow to switch between portrait and landscape or change the paper size mid document. But a macro can be written, which allows this. It's basically a changed copy of \newgeometry, called \newgeometryfull here. Use it with care! Additionally, \pdfpageheight and \pdfpagewidth must be set, so the PDF viewer also knows the paper size. This is done in the a3page environments.

I removed the line \headwidth=\textwidth, because \headwidth doesn't exist in your example. But may be you need it for headers not present here.

The result:

The code:

\documentclass[12pt]{article}

% Pictures & Labels
\usepackage[capitalise]{cleveref}
\usepackage{graphicx}
\usepackage{caption}
\usepackage{subcaption}
\usepackage{textcomp}
\usepackage{float}
\usepackage[section]{placeins}
% Paper Style
\usepackage[a4paper,
            width=150 mm,
            top=25 mm,
            bottom=25 mm]
            {geometry}
\usepackage[title,toc,page]{appendix}

\usepackage{lipsum}

\makeatletter
% like \newgeometry, but also allows change of landscape/portrait and paper size
% to be used with caution!
\newcommand{\newgeometryfull}[1]{%
  \clearpage
  \Gm@restore@org
  \Gm@initnewgm
%  \Gm@newgmtrue
  \setkeys{Gm}{#1}%
%  \Gm@newgmfalse
  \Gm@process
  \ifnum\mag=\@m\else\Gm@magtooffset\fi
  \Gm@changelayout
  \Gm@showparams{newgeometry}}%
\makeatother

%% Allow A3 sheets - - New environment
\newenvironment{a3page}{%
    \newgeometryfull{a3paper,landscape,width=360 mm,top=25 mm,bottom=25 mm}
    % set the correct dimension for the PDF viewer
    \pdfpageheight=\paperheight
    \pdfpagewidth=\paperwidth
}{%
    \restoregeometry
    % set the correct dimension for the PDF viewer
    \pdfpageheight=\paperheight
    \pdfpagewidth=\paperwidth
}

\begin{document}
\section{A test}
\lipsum[1]

%% Start A3 page
\begin{a3page}
\section{Title}
\lipsum[1]

\begin{figure}[H]
\centering
        \begin{subfigure}[t]{0.14\textwidth}
                \centering
                \includegraphics[height=4cm]{example-image-9x16.png}
                \caption{}
                \label{subfig:a}
        \end{subfigure}%
        %
        %
        \begin{subfigure}[t]{0.14\textwidth}
                \centering
                \includegraphics[height=4cm]{example-image-9x16.png}
                \caption{}
                \label{subfig:b}
        \end{subfigure}%
        %
        %
        \begin{subfigure}[t]{0.14\textwidth}
                \centering
                \includegraphics[height=4cm]{example-image-9x16.png}
                \caption{}
                \label{subfig:c}
        \end{subfigure}%
        %
        %
        \begin{subfigure}[t]{0.14\textwidth}
                \centering
                \includegraphics[height=4cm]{example-image-9x16.png}
                \caption{}
                \label{subfig:d}
        \end{subfigure}%
        %
        %
        \begin{subfigure}[t]{0.14\textwidth}
                \centering
                \includegraphics[height=4cm]{example-image-9x16.png}
                \caption{}
                \label{subfig:e}
        \end{subfigure}%
        %
        %
        \begin{subfigure}[t]{0.14\textwidth}
                \centering
                \includegraphics[height=4cm]{example-image-9x16.png}
                \caption{}
                \label{subfig:f}
        \end{subfigure}%
        %
        %
        \begin{subfigure}[t]{0.14\textwidth}
                \centering
                \includegraphics[height=4cm]{example-image-9x16.png}
                \caption{}
                \label{subfig:g}
        \end{subfigure}%
        %
        \caption{This caption isn't centered}\label{fig:}
\end{figure}

\lipsum[1]
\end{a3page}

\section{Another test}
\lipsum[1]

\end{document}