Generally PS1 games are whatever format the ripper decided to use. The games/console itself for most things just used a basic iso9660 (aka why you can put a PS1 CD in your computer and look at the files without needing another tool or specially hacked drive like a lot of other things) and thus can be edited accordingly. Some games, notably Square (Enix), did some raw reads of the disc outside the iso9660 stuff and why ripping audio for them can be a pain.
.bin is a generic name for a file extension, though specifically with CD rips of things then that tends to be bin and cue (did it come with a .cue file?). The PS1 in general is noted as being a huge mess here; when it was all still current there was no equivalent of https://scenerules.org/
and every CD burning/ripping program was in heavily unofficial competition over the PS1 market, and thus different groups/rippers would use any format that worked for their purposes. In the years since people have standardised on more but that then leaves out those using all those nice original Scene files, or indeed their own rips from back then (not much downloading when going to the rental shop would be 10x quicker -- I don't know if there is a piracy equivalent of "the latency sucks but never underestimate the bandwidth of a car boot full of hard drives" but yeah).
To that end most today will use one of the various iso handling tools, most of which are paid (wikipedia I know but it is a nice list https://en.wikipedia.org/wiki/Comparison_of_disc_image_software
, Ultraiso and magiciso probably being the things most are compared to) but there are some basic tools that do CD format which is nice even if you might have to extract the whole disc and rebuild it rather than having the tool overwrite it for you.
Alternative. If your edit has left the file the same size as the original, or smaller such that you can pad it out (bigger might work if there is some blank space for it to take up after the file originally finished but let's not go there), then overwriting with a hex editor is not the worst plan. Your extraction tool might tell you where in the iso the file came from, however it can be just as easy to take a big chunk of the original file (try to make it something hopefully at least a bit unique rather than the same 10 bytes that certain files all start with), open up a hex editor, tell it to search for that chunk of file and then note where it finds it, overwriting with your edit. Some iso formats might complain if the data is different if they have some kind of checking aspect but cross that bridge as and when.