Greek letters with Carlito font

Question

I need to typeset a few Greek letters in a document. I am using the sans serif font Carlito (provided by the package carlito) and babel to switch between Latin and Greek. There are Greek letters available in the font, but my Greek letters are still typeset with the default font.

How to get the Greek letters to use Carlito as well?

I am stuck with the classicthesis package for now, so I have to use pdfLaTeX (no LuaTeX or XeTeX).

\documentclass{article}

\usepackage[greek,english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[sfdefault,lf]{carlito}

\begin{document}

abmp
\greektext abmp

\end{document}

This is what I get:

Although these are avilable acoording to the font documentation (ftp://ftp.funet.fi/pub/TeX/CTAN/fonts/carlito/doc/Carlito-Regular.pdf):

UPDATE:

Here is the result that I am aiming (typeset with XeLaTeX):

\documentclass{article}

\usepackage{fontspec}
    \setmainfont{Carlito}

\begin{document}
abmpαβμπ
\end{document}

The suggestions by Steven B. Segletes got me this far, but the letters are still misaligned:

\documentclass{article}

\usepackage{graphicx,scalerel}
\usepackage[utf8]{inputenc}
\usepackage[sfdefault,lf]{carlito}

\newcommand\newalpha{\ThisStyle{%
  \setbox0=\hbox{$\SavedStyle\alpha$}\includegraphics[height=\ht0]{alpha}}}
\newcommand\newbeta{\ThisStyle{%
  \setbox0=\hbox{$\SavedStyle\beta$}\includegraphics[height=\ht0]{beta}}}
\newcommand\newmu{\ThisStyle{%
  \setbox0=\hbox{$\SavedStyle\mu$}\includegraphics[height=\ht0]{mu}}}
\newcommand\newpi{\ThisStyle{%
  \setbox0=\hbox{$\SavedStyle\pi$}\includegraphics[height=\ht0]{pi}}}

\begin{document}
abmp\newalpha\newbeta\newmu\newpi
\end{document}

Answer 1

So this is not the exact answer to my own question, but rather a workaround that I will be using if no better solution emerges.

I have been using the textgreek package to get the Greek letters in text mode. I discovered that it automatically switches to Symbol font if Helvetica is used (helvet), because the other fonts provided by textgreek do not go well with Sans Serif in general. So I ended up redefining the needed symbols to use Helvetica instead of Carlito (which effectively changed the font to Symbol), and I also resized them the match the glyph size in Carlito.

\documentclass{article}

\usepackage{graphicx}
\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}

\usepackage{textgreek}
\usepackage{helvet}
\usepackage[sfdefault,lf]{carlito}

\DeclareRobustCommand{\usehelv}[1]{{\fontfamily{phv}\selectfont\resizebox{!}{.92\height}{#1}}}
\let\oldalpha\textalpha
\renewcommand{\textalpha}{\usehelv{\oldalpha}}
\let\oldbeta\textbeta
\renewcommand{\textbeta}{\usehelv{\oldbeta}}
\let\oldpi\textpi
\renewcommand{\textpi}{\usehelv{\oldpi}}
\let\oldomega\textomega
\renewcommand{\textomega}{\usehelv{\oldomega}}

\begin{document}

abpo
\textalpha\textbeta\textpi\textomega

\end{document}