How to make text appear in the middle of a phantom in math mode?

You don't really need to use \mathclap, you can just place what you want to center in a box of the appropriate width.

\documentclass{article}
\usepackage{amsmath}
\usepackage{calc}
\newcommand*{\mask}[2]{%
    \mathord{\makebox[\widthof{\(#1\)}]{\(#2\)}}%
}
\begin{document}
\begin{gather*}
    1 + \mask{ABC}{B} + 2 \\
    1 + ABC + 2
\end{gather*}
\end{document}

A version that works in all styles.

\documentclass{article}
\usepackage{amsmath}

\makeatletter
\newcommand{\mask}[2]{{\mathpalette\mask@{{#1}{#2}}}}
\newcommand{\mask@}[2]{\mask@@{#1}#2}
\newcommand{\mask@@}[3]{%
  \settowidth{\dimen@}{$\m@th#1#2$}%
  \makebox[\dimen@]{$\m@th#1#3$}%
}
\makeatother

\begin{document}

\begin{gather*}
1 + \mask{ABC}{B} + 2 \\
1 + ABC + 2 \\
1 + \mask{\int_0^1 f(x)\,dx}{y}+2 \\
1 + \int_0^1 f(x)\,dx+2
\end{gather*}

\[
\begin{array}{c}
1 + \mask{\int_0^1 f(x)\,dx}{y}+2 \\
1 + \int_0^1 f(x)\,dx+2
\end{array}
\]

\[
\sum_{\substack{1\le \mask{w}{i}\le n \\ 1\le w\le n}} a_{iw}
\]

\end{document}