Your datestyle was inadequate for the narrow date band provided. So I created an alternate (vertical) date style (named \mychronovdatestyle, note the v) for such narrowly spaced events.
I also added a number of % chars at line end, to suppress stray spaces from getting introduced into the labels.
\documentclass{article}
\usepackage{chronosys}
\newcommand{\mychronodatestyle}[1]{%
\pgfmathparse{equal(sign(#1),-1)? int(abs(#1)):#1 }%
\pgfmathresult%
\pgfmathparse{equal(sign(#1),-1)? "BC":}%
~\pgfmathresult%
}
\newcommand{\mychronovdatestyle}[1]{%
\pgfmathparse{equal(sign(#1),-1)? int(abs(#1)):#1 }%
\edef\tmp{\pgfmathresult}%
\pgfmathparse{equal(sign(#1),-1)? "BC":}%
\kern-1.5pt\rotatebox[origin=left]{90}{--- \tmp~\pgfmathresult}%
}
\catcode`\!=11
\def\eventyear{\!chreventyear}
\catcode`\!=12
\newcommand{\myeventdatestyle}[1]{%
\pgfmathparse{equal(sign(\eventyear),-1)? int(abs(\eventyear)):"#1"}%
\pgfmathresult%
\pgfmathparse{equal(sign(\eventyear),-1)? "BC":}%
~\pgfmathresult%
}
\begin{document}
\startchronology
[startyear=-800,stopyear=-14, arrow=false, height=0.2em, dateselevation=10pt,
datesstyle=\mychronodatestyle
]
\chronoevent[datestyle=\myeventdatestyle ]
{12/-753}{Rome's foundation}
\chronoperiode[color=red,datesstyle=\mychronovdatestyle,dateselevation=-7pt]
{-350}{-320}{important event}
\stopchronology
\end{document}

SUPPLEMENT
The OP asks if the vertical dates could be placed below the line, at the expense of omitting the label.
I took no effort to make it pretty, but just minimized the modifications necessary from the above MWE:
removed "important event" text, leaving argument blank.
changed the dateselevation from -7pt to -10pt.
changed the \rotatebox angle from 90 to -90.
changed the label pre-kern from -1.5pt to 1.5pt.
The new MWE:
\documentclass{article}
\usepackage{chronosys}
\newcommand{\mychronodatestyle}[1]{%
\pgfmathparse{equal(sign(#1),-1)? int(abs(#1)):#1 }%
\pgfmathresult%
\pgfmathparse{equal(sign(#1),-1)? "BC":}%
~\pgfmathresult%
}
\newcommand{\mychronovdatestyle}[1]{%
\pgfmathparse{equal(sign(#1),-1)? int(abs(#1)):#1 }%
\edef\tmp{\pgfmathresult}%
\pgfmathparse{equal(sign(#1),-1)? "BC":}%
\kern1.5pt\rotatebox[origin=left]{-90}{--- \tmp~\pgfmathresult}%
}
\catcode`\!=11
\def\eventyear{\!chreventyear}
\catcode`\!=12
\newcommand{\myeventdatestyle}[1]{%
\pgfmathparse{equal(sign(\eventyear),-1)? int(abs(\eventyear)):"#1"}%
\pgfmathresult%
\pgfmathparse{equal(sign(\eventyear),-1)? "BC":}%
~\pgfmathresult%
}
\begin{document}
\startchronology
[startyear=-800,stopyear=-14, arrow=false, height=0.2em, dateselevation=10pt,
datesstyle=\mychronodatestyle
]
\chronoevent[datestyle=\myeventdatestyle ]
{12/-753}{Rome's foundation}
\chronoperiode[color=red,datesstyle=\mychronovdatestyle,dateselevation=-10pt]
{-350}{-320}{}
\stopchronology
\end{document}
