I'll vouch for the PSV as the way to go in this case, though the PS1 games I still play are mostly RPGs. Never noticed latency but with that genre it's not really a problem if it does happen, and playing on the go / suspending at any time is a big factor for me.
Hyperkin HDTV 720p Cables for PlayStation
Playing PS1 games on the other Sony platforms doesn't really work for me, got other games I'd rather play on them and the older graphics look better on a smaller screen.
No experience with emulation besides ePSXe on PC a few years ago, and that was very hit-and-miss back then. Can't comment on the current situation, though savestates were a godsent.