Is there a variant of the digit 1 with a tighter, more "within-text-like" bounding box?
It sounds like you're looking for proportionally spaced, rather than the usual fixed spaced, numerals.
Can I easily build one?
This depends on which font and which LaTeX compiler you employ. Your mission is fairly easy to accomplish if you employ either XeLaTeX or LuaLaTeX, rather than the default pdfLaTeX, to compile your documents. Compare the first and second tikz pictures below; the effect of switching to a proportionally-spaced numeral 1 is definitely visible, but it is quite subtle.
For maximum visual effect, you may actually want to switch to so-called "oldstyle" numerals; see the third tikz picture below.

% !TEX TS-program = lualatex
% or: !TEX TS-program = xelatex
\documentclass{article}
\usepackage{fontspec}
% Set the default font (fixed-width lining-type numerals):
\setmainfont{Latin Modern Roman}
% Same font family, but with proportionally-spaced numerals:
\newfontfamily\LMPropNums{Latin Modern Roman}[Numbers={Proportional}]
% Same font family, but with oldstyle numerals:
\newfontfamily\LMOldStyle{Latin Modern Roman}[Numbers=OldStyle]
\newcommand\POne{{\LMPropNums 1}} % just the numeral "1"
\usepackage{tikz}
\usepackage{siunitx}
\begin{document}
\begin{tabular}{@{}ll@{}}
111 & fixed spaced \
\LMPropNums 111 & proportionally spaced \
000 & fixed spaced \
\LMPropNums 000 & proportionally spaced
\end{tabular}
\begin{tikzpicture}
\draw (0, 0)
node (image){\includegraphics[width=1.1cm]{example-image} };
\draw[red, x={(image.south east)}, y={(image.north west)}] (0.5, 0.5)
node (text){\SI{100}{\micro\meter} };
\end{tikzpicture}
% Use a proportionally-spaced digit "1":
\begin{tikzpicture}
\draw (0, 0)
node (image){\includegraphics[width=1.1cm]{example-image} };
\draw[red, x={(image.south east)}, y={(image.north west)}] (0.5, 0.5)
node (text){\POne00,\si{\micro\meter} };
\end{tikzpicture}
% Switch to old-style numerals:
{\LMOldStyle
\begin{tikzpicture}
\draw (0, 0)
node (image){\includegraphics[width=1.1cm]{example-image} };
\draw[red, x={(image.south east)}, y={(image.north west)}] (0.5, 0.5)
node (text){100,\si{\micro\meter} };
\end{tikzpicture}}
\end{document}
1. Please tell us which text font and math fonts you use. – Mico Nov 07 '20 at 06:20