I'm not sure how the statute of limitations has any applicability to your case whatsoever since you state you already plead guilty and have been sentenced to 1 year in county jail.
Example of statute of limitations being applicable: You committed the crime of pandering 3 years ago. No charges were filed within 3 years. Charges were filed 3 1/2 years after the felony offense was allegedly committed. You then have the right to file a motion for dismissal based on the statute of limitations.
So, unless there is something I am misunderstanding, it doesn't apply to your case.
As to a request to do your time in LA County jail: My first thought is - did you/your attorney request that you could serve your time in Los Angeles county when you were sentenced? If not, then you could "walk in" to court asap, by going to the clerks' office at 8am , requesting that your matter be heard, then going to the particular Courtrooom. Then when the Judge calls your matter, make your special request. It's up to the Judge. I recommend contacting the attorney who handled this matter - you may want him/her to help you with this. And he/she may have a good idea as to whether the Judge will grant this request.
I imagine you also want your Felony Formal Probation to be in LA County as opposed to SBD County since you state you live in LA County. You can request that from the Judge as well. It's up to the Judge. If the Judge denies this request, you will have to work your SBD Probation Officer to have your Probation transferred to LA County. That can take some time for the paperwork to go through.