I think you are going to have to serve the jail time, and that you should do it immediately to avoid getting arrested out of the blue one day while you are driving your kids to something fun. The jail sentence is not going to go away.
However, a person who has an attorney's representation in this situation (either a private attorney that you hire or an assigned public defender) can instruct that attorney to file a Motion for Modification of Sentence.
Speaking from the court's perspective, a jail sentence is punishment and involuntary loss of liberty. You do not get to do it on your terms - from the court's perspective.
You need to get on this immediately.
The best way to handle the motion would be to send the attorney in for you, and you do not go at first to avoid the possibility of being taken into custody right away. Then if the court will not rule on the motion without you there (which is a possibility) you would have to go in person, but that appearance could be scheduled in a way so that if they do remand you instantly, you will have had a couple of days to prepare.
Compromise is not really an activity I would expect a criminal court judge to participate in. The judge and the court don't have any motivation to negotiate with you. They can just wait until you are arrested and brought in.
Usually it is not the jailer that issues a warrant. I do not know which court you are in, and am speaking generally. The way it usually works is that the jail will report to the court that the defendant did not serve the time, and then the court will issue the arrest warrant, not the jailer. Jailers do not have power to issue warrants. Warrants are court orders.
If you are on probation also, it is possible that this problem could cause a revoking of your probation.
One thing I know for 100% is that it will not magically go away.
Another thing I know for sure is that once the matter is before the court, the judge is going to want to know why you did not address the problem sooner.
So for those reasons, I'd say you must contact the court clerk now, to determine the status of your case as of now.
If you are on probation, and the court clerk tells you there is no warrant yet, you can contact the court's probation department and ask for instructions.
At the same time, a person in this position should get an attorney working on that motion so that it can be filed later today.
Those are my thoughts.