How to generate a new reference identifier by citing the same bib entry, with a twist

This is not a complete solution but I hope that it will give you the basis of one. To tweak the formatting (e.g., if you want just first initials or whatever), see the biblatex manual.

\documentclass{article}

\usepackage{lipsum}
\usepackage{filecontents}
\usepackage{endnotes}
\usepackage[citestyle=verbose-note, notetype=endonly, singletitle=false, backend=biber]{biblatex}    
% I haven't tested with BibTeX, so I don't want this code uncommented here. However, I gather from an edit that Serguei has determined it to work with the BibTeX backend. To use this, comment the previous line and uncomment the next. Note, however, that some features are not supported by the legacy backend.  
% \usepackage[citestyle=verbose-note, notetype=endonly, singletitle=false, backend=bibtex]{biblatex}

\bibliography{\jobname}
\begin{filecontents}{\jobname.bib}
    @misc
    {
        death-star,
        author       = {Bevel Lemelisk and Wilhuff Tarkin and Darth Vader and Darth Sidious},
        title        = {{Death Star}},
        howpublished = {Alderaan and Yavin 4},
        year         = {0 BBY}
    }
    @misc
    {
        death-star-2,
        author       = {Bevel Lemelisk and Wilhuff Tarkin and Darth Vader and Darth Sidious},
        title        = {{Death Star II}},
        howpublished = {Endor},
        year         = {4 ABY}
    }
\end{filecontents}

\def\enotesize{\normalsize}
\renewcommand{\notesname}{References}
\renewcommand{\makeenmark}{\mkbibbrackets{\theenmark}}

\makeatletter
\renewbibmacro*{footcite:note}{% adapted from verbose-note.cbx
    \ifnameundef{labelname}
    {\printfield{label}}{}%
    \setunit*{\addspace}%
    \printtext{%
        \mkbibbrackets{%
            \ref{cbx@\csuse{cbx@f@\thefield{entrykey}}}}%
        \iftoggle{cbx:pageref}
        {\ifsamepage{\the\value{instcount}}
            {\csuse{cbx@f@\thefield{entrykey}}}
            {}
            {\addcomma\space\bibstring{page}\addnbspace
                \pageref{cbx@\csuse{cbx@f@\thefield{entrykey}}}}}
        {}}}
\makeatother

\begin{document}

    They first built \emph{Death Star}~\autocite{death-star}.

    \lipsum[1]

    The design flaw was found in \autocite[Lemelisk et al., Chapter 3, p. 123][]{death-star}.

    \lipsum[2]

    To address the flaw, they designed \emph{Death Star 2}~\autocite{death-star-2}
    that featured many smaller diameter heat exhaust vents.

    \renewcommand{\makeenmark}{\mkbibbrackets{\theenmark}\addspace}
    \theendnotes

\end{document}

EDIT [by Serguei]

• A GitHub style project for leonardo.cls was started and used in a couple of article submissions based on this answer. Further contributions are welcome.