Ok but the PLA is an alliance, I presume you are a member? and therefore are a committee run non for profit organisation with charitable status.
I am not very familiar with the PLA however I am a member of associations and they seem similar in that they offer support, advice, and cheap insurance packages. All very nice and handy should you need it, but not necessary for Ofsted registration.
You do not need to be a member of the PLA to operate as a committee run charitable group.
As far as I can ascertain from the registration process, and I have looked at it a lot for varying reasons, the process is generally similar regardless of your operating status.
I bought a nursery a few years ago and it went like this.
Because I purchased the company (an existing entity, retained the name and company number) I did not need to be re-inspected and could remain open throughout process, I was required to have a suitable person interview but operated as manager whilst awaiting this.
If I had bought the company and changed the name and / or company number I would have been required to re-register and have another full inspection, however as the building was registered could have operated whilst awaiting inspection.
From the rules on Committee run groups ofsted carry out checks on each committee member, and when one resign, ofsted are notified, when another is appointed again ofsted carry out another suitable person check. If all members resign at the same time which may happen at the agm or if the group closes the ofsted deem that the provision is closed, therefore not operating. What I am not sure about is how many members are needed on the committee, I am presuming that it would be more than 1, but have not found a definitive number for operating. I would suggest that if you want to remain a committee that you check with the PLA they should have specific info as it's their field iyswim.
the factsheet is here www.ofsted.gov.uk/resources/factsheet-childcare-committee-run-registered-childcare-provision
If you are a member of the committee already then if other members resign (as sounds the case as you say you have staffing issues) I suggest that you find another member to form a committee with.
Ultimately I think the advice your LA has given would be the best bet, if you are a member of the committee currently then I would suggest that as you already have a premises registered and as you have already had a suitable person check as per the committee registration procedures their may be a case for you setting up a new name as the named person without a gap in the changeover, and running the pre-school as a business or charity if you prefer without being a PLA member (although I still don't know where the 6 months comes from, I understood the ethos of PLA was for accesible care for all with this in mind I doubt they would advocate a 6 month gap)
If you want to set up as childcare on non-domestic premises the fact sheet is here
www.ofsted.gov.uk/resources/guide-registration-early-years-register-childcare-provider-domestic-or-non-domestic-premises.