You can define your own variation of the commands.
\documentclass{amsart}
\usepackage{listofitems}
\ExplSyntaxOn
\NewDocumentCommand{\readnamedlist}{smm}
{
\IfBooleanTF{#1}
{\exp_args:NNc \readlist*{#2}{#3}} % * variant
{\exp_args:Nc \readlist{#2}{#3}}
}
\NewDocumentCommand{\greadnamedlist}{smm}
{
\IfBooleanTF{#1}
{\exp_args:NNc \greadlist*{#2}{#3}} % *-variant
{\exp_args:Nc \greadlist{#2}{#3}}
}
\ExplSyntaxOff
\begin{document}
\newcounter{num}\setcounter{num}{1}
\setsepchar{,}
\greadnamedlist*{LIST\roman{num}}{a,b,c}
\LISTi[1],\LISTi[2],\LISTi[3]
\stepcounter{num}
\readnamedlist{LIST\roman{num}}{d,e,f}
\LISTii[1],\LISTii[2],\LISTii[3]
\end{document}

What does \exp_args:Nc
do? It jumps over the next token and applies \csname...\endcsname
to the contents of the following braced group. Similarly \exp_args:NNc
that jumps over two tokens.
If you prefer a “classical” implementation, here's an argument reversing procedure.
\documentclass{amsart}
\usepackage{listofitems}
\makeatletter
\newcommand{\readnamedlist}{%
\@ifstar{\@readnamedlist{\readlist*}}{\@readnamedlist{\readlist}}%
}
\newcommand{\greadnamedlist}{%
\@ifstar{\@readnamedlist{\greadlist*}}{\@readnamedlist{\greadlist}}%
}
\newcommand{\@readnamedlist}[2]{%
\expandafter\@@readnamedlist\csname #2\endcsname{#1}%
}
\newcommand{\@@readnamedlist}[2]{#2#1}
\makeatother
\begin{document}
\newcounter{num}\setcounter{num}{1}
\setsepchar{,}
\greadnamedlist*{LIST\roman{num}}{a,b,c}
\LISTi[1],\LISTi[2],\LISTi[3]
\stepcounter{num}
\readnamedlist{LIST\roman{num}}{d,e,f}
\LISTii[1],\LISTii[2],\LISTii[3]
\end{document}