With the help of @michal.h21's comments, I was able to reach a solution to this.
My initial attempt had been to get the relevant list configuration settings from the tex4ht folder, and adapt them to my purposes in my own config file. But I could then not build my document.
However, as Michal taught me, those files commonly use : in macros names, which cannot be ported without due care. With that information in hand, I could then reach the desired results with the following settings:
A build file filename.mk4:
set_settings{tex4ht_sty_par = "filename," .. settings.tex4ht_sty_par}
settings_add{ tex4ht_sty_par = ",NoFonts,-css" }
Make:latexmk()
A config file filename.cfg:
\Preamble{html5}
\Configure{emph}{\ifvmode\ShowPar\fi\HCode{<em>}}{\HCode{</em>}}
\Configure{textbf}{\ifvmode\ShowPar\fi\HCode{<b>}}{\HCode{</b>}}
\Configure{HtmlPar}
{\EndP\Tg<p>}
{\EndP\Tg<p>}
{\HCode{</p>\Hnewline}}
{\HCode{</p>\Hnewline}}
\catcode`\:=11
\ConfigureList{enumerate}%
{\EndP\HCode{<dl \a:LRdir class="enumerate-enumitem">}%
\PushMacro\end:itm
\global\let\end:itm=\empty}
{\PopMacro\end:itm \global\let\end:itm \end:itm
\EndP\HCode{</dd></dl>}\ShowPar}
{\end:itm \global\def\end:itm{\EndP\Tg</dd>}\HCode{<dt
class="enumerate-enumitem" style="float:left; clear:left; margin-left:1em; margin-right:1em; padding-top:5px;">}\bgroup \bf}
{\egroup\EndP\HCode{</dt><dd\Hnewline class="enumerate-enumitem" style="padding-top:5px;">}}
\catcode`\:=12
\begin{document}
\EndPreamble
(Note the definition of \ConfigureList{enumerate} was taken from html4.4ht and only the style settings were added, directly within the tags of interest).
The file itself filename.tex:
\documentclass{article}
\usepackage{enumitem}
\setlist[itemize]{leftmargin=4em, itemsep=0pt}
\setlist[enumerate,1]{wide, labelindent=0pt, label={Part \Roman* --}}
\setlist[enumerate,2]{wide=1em, label={\arabic* --}, resume}
\newlist{biblio}{itemize}{1}
\setlist[biblio]{label=\textbullet}
\begin{document}
\begin{enumerate}
\item A first item
\begin{enumerate}
\item A first subitem in enumerate
\item A second one
\begin{itemize}
\item Itemize item
\end{itemize}
\end{enumerate}
\end{enumerate}
\end{document}
Built with make4ht -uf html5+tidy filename.tex will result in:

From the following html code:
<body>
<dl class="enumerate-enumitem">
<dt class="enumerate-enumitem" style="float:left; clear:left; margin-left:1em; margin-right:1em; padding-top:5px;">Part I –</dt>
<dd class="enumerate-enumitem" style="padding-top:5px;">A first item
<dl class="enumerate-enumitem">
<dt class="enumerate-enumitem" style="float:left; clear:left; margin-left:1em; margin-right:1em; padding-top:5px;">1 –</dt>
<dd class="enumerate-enumitem" style="padding-top:5px;">A first subitem in enumerate</dd>
<dt class="enumerate-enumitem" style="float:left; clear:left; margin-left:1em; margin-right:1em; padding-top:5px;">2 –</dt>
<dd class="enumerate-enumitem" style="padding-top:5px;">A second one
<ul class="itemize1">
<li class="itemize">Itemize item</li>
</ul>
</dd>
</dl>
</dd>
</dl>
</body>
Which can now be properly copy-pasted in a blog post in Wordpress (for it is both css-less and headless).
Previous answer
This works to produce a no-css html, but is not yet good enough, because the style settings are set in the html file head, so when copying to a wordpress blog post these settings were lost. But I keep it here for this might be useful for other purposes (and, strictly speaking, it does answer the OP).
I think I managed a reasonable solution to this (though in not a very useful form for my intended purpose). In no way I could tweak the html tags to get a description list to work without some styling (it doesn't mean it can't be done...). I also could not get the tex4ht option css-in to actually embed the style sheet in my html file. So I had somehow to bring the style manually to my file. I built the document with defaults letting it thus give me a css style sheet, where I found:
dt.enumerate-enumitem{float:left; clear:left; margin-left:1em; margin-right:1em;}
So, I added to my config file:
\Configure{@HEAD}{\HCode{<style>
dt.enumerate-enumitem{float:left; clear:left; margin-left:1em; margin-right:1em;}
</style>}}
And then ran things with option -css. The result is satisfactory:

In detail, my settings for this are:
A build file filename.mk4
set_settings{tex4ht_sty_par = "filename," .. settings.tex4ht_sty_par}
settings_add{ tex4ht_sty_par = ",NoFonts,-css" }
Make:latexmk()
A config file filename.cfg:
\Preamble{html5}
\Configure{emph}{\ifvmode\ShowPar\fi\HCode{<em>}}{\HCode{</em>}}
\Configure{textbf}{\ifvmode\ShowPar\fi\HCode{<b>}}{\HCode{</b>}}
\Configure{HtmlPar}
{\EndP\Tg<p>}
{\EndP\Tg<p>}
{\HCode{</p>\Hnewline}}
{\HCode{</p>\Hnewline}}
\Configure{@HEAD}{\HCode{<style>
dt.enumerate-enumitem{float:left; clear:left; margin-left:1em; margin-right:1em;}
</style>}}
\begin{document}
\EndPreamble
The file itself filename.tex:
\documentclass{article}
\usepackage{enumitem}
\setlist[itemize]{leftmargin=4em, itemsep=0pt}
\setlist[enumerate,1]{wide, labelindent=0pt, label={Part \Roman* --}}
\setlist[enumerate,2]{wide=1em, label={\arabic* --}, resume}
\begin{document}
\begin{enumerate}
\item A first item
\begin{enumerate}
\item A first subitem in enumerate
\item A second one
\begin{itemize}
\item Itemize item
\end{itemize}
\end{enumerate}
\end{enumerate}
\end{document}
Built with:
make4ht -uf html5+tidy filename.tex
<head>is trimmed in the process... – gusbrs Feb 20 '19 at 15:02<dt>tag (I've tested a manual change). But I can't seem to convincetex4htto do so. – gusbrs Feb 20 '19 at 17:22\ConfigureListdefinitions inside the tex4ht directory. I tried to copy some of them to my config file to adjust them. But usually, when I do so, the build process fails. What's the catch? – gusbrs Feb 20 '19 at 17:25.4htfiles contains commands with:as a part of their names. this can cause compilation errors as this character is not normally part of command names. – michal.h21 Feb 20 '19 at 17:31:defined? And what is their scope? – gusbrs Feb 20 '19 at 17:334htfiles enable:in command names. they can be used with\csname ...\endcsnamein normal TeX code. – michal.h21 Feb 20 '19 at 17:35\DeleteMark, which I think standard definitions contain – michal.h21 Feb 20 '19 at 17:35:. Thanks! The general procedure would then be to copy those definitions and put any such commands between\csname ...\endcsname? Or is there a smarter/better way? – gusbrs Feb 20 '19 at 17:38:to letter at the begin of the .cfg file, just don't forget to change it back to "other" at the end – michal.h21 Feb 20 '19 at 17:56