midway label for edge

You're almost there with the TEST2 node, you just need to move the node to before the last coordinate. That is, instead of

(a) edge (b) node {foo}

you do

(a) edge  node {foo} (b)

If you want the label parallel to the edge, use

(a) edge  node[sloped,above] {foo} (b)

There is also a shorter syntax available if you add \usetikzlibrary{quotes}, and that is

(a) edge["foo"sloped] (b)

Using the first syntax in your code:

\documentclass[border=10pt]{standalone}
%%%%%%%%%%%%%%%%%% PACKAGE %%%%%%%%%%%%%%%%%%
\usepackage{tikz, tkz-euclide}%  permet de dessiner des figures, des graphiques
\usepackage{adjustbox}% permet de déterminer une taille de fenêtre
%%  FONT
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{tgadventor}% paquet de police de caractère TGadventor
\usepackage{sansmath}%  Copie-colle la police active dans 
%                       \sfdefault (/!\ N'EST PAS UNE POLICE DE CARACTÈRES)
\usepackage{xcolor}
%%%%%%%%%%%%%%%%%% INPUT %%%%%%%%%%%%%%%%%%
%\input{preamble.tex}
%\input{parameters.tex}

%\input{types/f2d_figlab}
%%%%%%%%%%%%%%%%%% SETUP %%%%%%%%%%%%%%%%%%
\tikzset{fig2D/.style={font={\sansmath\sffamily\Large}, line width=0.4mm, line cap=round, line join=round, >=latex, x=1.0cm, y=1.0cm,}}

%%%%%%%%%%%%%%%%%%%%%%%% DOTTED TRIANGLOID %%%%%%%%%%%%%%%%%%%%%%%%
\tikzset{pics/dotted triangloid/.style={code={%
    \tikzset{dotted triangloid/.cd,#1}%
    \def\pv##1{\pgfkeysvalueof{/tikz/dotted triangloid/##1}}
    \draw [solid, pic actions, fill=\pv{fill}]
    %%  point o - bas gauche
    (0,0) coordinate (o)
    node [circle, fill=black, scale=0.5, label={[below, inner sep=\pv{inner sep}, shift={(-2pt,-6pt)}]\pv{labo}}] {}
    %% point a - bas droite
    (\pv{scale}*\pv{width},0) coordinate (a)
    node [circle, fill=black, scale=0.5, label={[below, inner sep=\pv{inner sep}, shift={(0pt,-6pt)}]\pv{laba}}] {}
    %% point b - sommet
    ({\pv{scale}*\pv{offset})},\pv{scale}*\pv{height}) coordinate (b)
    node [circle, fill=black, scale=0.5, label={[above, inner sep=\pv{inner sep},]\pv{labb}}] {}
    %% jonction entre les points
     (o) 
     -- (a) 
     node [midway, label={[below, inner sep=\pv{inner sep},shift={(0pt,-6pt)}]\pv{labA}}] {}
     -- (b)
     node [midway, label={[above right, inner sep=\pv{inner sep},]\pv{labB}}] {}
     -- (o)
     node [midway, label={[above left, inner sep=\pv{inner sep},,shift={(-2pt,0pt)}]\pv{labC}}] {}
     ;
  }},
  dotted triangloid/.cd,
  width/.initial=5,
  height/.initial=2,
  offset/.initial=0,
  labA/.initial=,
  labB/.initial=,
  labC/.initial=,
  labo/.initial=,
  laba/.initial=,
  labb/.initial=,
  inner sep/.initial=0pt,
  fill/.initial=black!0,
  scale/.initial=1,
}
%%%%%%%%%%%%%%%%%% DOCUMENT %%%%%%%%%%%%%%%%%%
\begin{document}
\begin{tikzpicture}[fig2D]

\pic {dotted triangloid={width=8.82, height=6.47, offset=2.35, labo=B, laba=C, labb=A, fill= orange!20}};

\path
    (o) -- (b) node [circle, pos=0.333, fill=black, scale=0.5, label=left: D] (d) {}
    (a) -- (b) node [circle, pos=0.333, fill=black, scale=0.5, label=above right: E] (e) {}
;
\draw
    (d) -- (e)
    (o) +(-6pt, 0) coordinate (o')
    (b) +(-6pt, 0) coordinate (b')
    (b) +(6pt, 0) coordinate (b'')
    (d) +(-6pt, -3pt) coordinate (d')
    (d) +(-6pt, 3pt) coordinate (d'')
    (o') edge[<->, densely dashed] node [sloped,above] {bar} (d') 
    (d'') edge[<->, densely dashed] node [sloped,above] {foo} (b')
;

\end{tikzpicture}
\end{document}

Compile with Asymptote.

size(300);
unitsize(1cm);

pair A=(0,0),B=4*dir(80),C=(5,0);
pair D=A+0.3*(B-A),E=C+0.3*(B-C);

real distance=.3;
pair D1=D+distance*dir(degrees(B-D)+90),
     B1=B+distance*dir(degrees(B-D)+90),
     D2=D+.2*dir(degrees(B-D)+90),
     A1=A+.2*dir(degrees(D-A)+90);
draw(A--B--C--cycle);
draw("TEST2",D--E);
draw(rotate(degrees(B-D))*Label("TEST1",align=(0,0),UnFill),B1--D1,dashed,Arrows());
draw(rotate(degrees(D-A))*"TEST3",D2--A1,dashed,Arrows());
// OR
// draw(Label("TEST1",Rotate(-dir(B1--D1)),align=(0,0),UnFill),B1--D1,dashed,Arrows());
// draw(Label("TEST3",Rotate(dir(A1--D2))),D2--A1,dashed,Arrows());
label("$A$",A,dir(-135));
label("$B$",B,dir(90));
label("$C$",C,dir(-45));
dot(D^^E);