I know you're not going to like this, but I would suggest you recycle the old pack and buy a new one.
LION batteries are not all that safe. Think of what happens when you leave Lithium out in the open on a humid day. What about when you pack the batteries back into the case? There is probably a thermal sensor that is used to signal end of charge. If you forget that, you might have a dangerous situation literally sitting in your lap. With regards to the charging circuit which might contain a processor and non-volatile memory - what usually happens is that the processor will track the age of the battery and report on the charge remaining accordingly. It's not a sure thing that swapping in new LION will not work. In fact I suspect it will. But it might take a few charge cycles until you see the "new" expected life behavior. As for discharging LION. Never do this. This is why you seldom see LION by themselves. They are almost always in a pack where special circuits shutdown the battery before they are completely discharged. Such is how I would expect your laptop battery pack to behave.
BTW, it's probably not the computer telling you how much time you have left to run your computer. It's probably the processor inside the battery pack. That's why there are so many terminals. Some are for power. Some are (probably) for safety. And some are (probably) so the computer can talk to the processor inside the pack. I think if you wikipedia SBus you can find out more about this.
BTW, I might try rebuilding a pack (I've never done it). But I would never take such a pack into a closed space like an airplane.
Edit: added later
A side note: I hope you purchased cells with solder-tabs pre-attached, or you're going to have one hell of a time trying to connect all your cells together!
I'm sure UAir knows this, you'll need a spot welder. We had one at my old work. Not much to them - but very specialized as for their intended purpose. Do not subject the LION cell to anything out of the ordinary! Like trying to solder wire to them! At best - if they have an internal thermal fuse - they will stop working. At worst - well, recall what I said about Lithium on a humid day.