Enforcing Full Stop at the End of Hanging Captions within `xltabular`

Question

The following posts provides a solution for adding a full stop at the end of hanging captions, but I'm encountering issues when applying this in the context of xltabular.

Can I extend this functionality to also work for captions in xltabular? I would love to get help!

Below is a minimal working example (MWE). It demonstrates the successful implementation of adding a full stop to hanging captions but also illustrates the issue with xltabular table captions where the full stop is not being added.

\documentclass{article}
\usepackage{amsthm} % Adds a full stop at the end of each caption automatically (if there is none)
\usepackage{caption}   % Use 'caption' Package
\usepackage{xltabular} % Load 'xltabular' package
\makeatletter
% Custom caption format based on the original hang format with "ABC" appended
\DeclareCaptionFormat{customhang}{%
  \caption@iflabelseparatorwithnewline
    {\caption@Error{%
       The option labelsep=\caption@labelsep@name' does not work\MessageBreak withformat=customhang'}}%
    {@hangfrom{#1#2}%
     \advance\caption@parindent\hangindent\relax
     \advance\caption@hangindent\hangindent\relax
     \caption@@par#3}}
\NewDocumentCommand{\CUSTOMcaption}{sO{#3}m}{%
  \IfBooleanTF{#1}{%
    \CAPTIONcaption*[#2@addpunct{.}]{#3@addpunct{.}}%
  }{%
    \CAPTIONcaption[#2@addpunct{.}]{#3@addpunct{.}}%
  }%
}
\makeatother
\captionsetup{format=customhang}
\AtBeginDocument{%
  \NewCommandCopy\CAPTIONcaption\caption
  \RenewCommandCopy\caption\CUSTOMcaption
}
\begin{document}
% ACTUAL ISSUE OCCURS HERE
\begin{xltabular}{\linewidth}{X}
    DEMO \
    \caption{Samle with xltabular} % Here a full stop should be added, but currently is not being added!
\end{xltabular}
% Just to ensure that the new solution also works with normal captions inside figures:
\begin{figure}
    \centering
    \caption{This is an example caption}
    \caption{Another sample with a very long caption and we hope that it will properly break into the next line with proper hanging...}
\end{figure}
\end{document}