A mark belongs to one sheet of paper
The slip is not issued to you or to your case. It is fixed to a single sheet and vouches for the signature and seal printed on that sheet. Everything else follows from this. The question at the start of the job is not what to prove but which piece of paper you are willing to put in an envelope.
Libya sits outside the Convention
Libya has not joined the Hague Apostille Convention. A British mark is still the first step, because the consular desk builds on it, but the file then goes on to the staff acting for Libya here for a second stamp. Two desks mean two trips through the post, and each trip is a chance for an envelope to go astray.
Where a replaceable version comes from
Civil records are the easy case. Westminster Register Office prints certified copies of birth, marriage and death entries, and the entry itself stays in the register whatever happens to the paper. Court papers work the same way through the court that issued them, and a public body will usually reissue what it signed.
An apostille from United Kingdom to Libya is worth planning around this point rather than around the calendar. Order the fresh copy first, check the spelling of every name on it while it is still cheap to fix, and only then start the journey through the two desks.
Handling the bundle on the way back
The British office works remotely from one address and returns files the way they arrived, so plan for post in both directions. Nothing on this route happens at a counter in the capital, and no local desk holds your file for collection while you are nearby.
In short, what you need for an apostille for Libya is a signed public version of each paper, a charge paid per paper, and the discipline to keep the marked bundle intact. The papers that cause trouble later are the ones that were irreplaceable when they set out.