None of the excuses you mention have any real effect on guilt or innocence. And the fine is a very small part of the total obligation you would have to pay if you are found (or plead) guilty. So, there may not be much wiggle room for a court other than to allow you to pay over a short period of time.
Here is the section:
22450. (a) The driver of any vehicle approaching a stop sign at the
entrance to, or within, an intersection shall stop at a limit line,
if marked, otherwise before entering the crosswalk on the near side
of the intersection.
If there is no limit line or crosswalk, the driver shall stop at
the entrance to the intersecting roadway.
By your own admission, you were on or over the limit line. A judge MIGHT decide that was sufficient ... maybe ... it depends on what the officer has to say.
if you take it to trial you may not have the option of traffic school if you lose. But, you can always try a trial by written declaration first and then see what happens.