I really don't understand what on earth you're on about. What on earth does the Falklands have to do with what options may exist for keeping the Ireland - Ulster border checkpoint free?
Gibraltar is no parallel at all. It has a checkpoint. There were claims earlier that Gibraltar's economy depended either on workers from Spain working in Gibraltar, or on workers from Gibraltar working in Spain. I'd have guessed it depended more on being a tax haven, but even so, it's yet another non-problem - just give those people work permits.
Let me try again.
It's agreed we all want to keep the Irish Border open.
In a no or poor-deal senario we will have no choice but to make it hard to check goods and people to maintain the integrity of the Single Market which we won't be part of.
Which could re-ignight the troubles which we don't want;
So the solution is to remain inside the Customs Union and Single Market which the government won't do - or have an arrangement which is remarkably similar which the EU say we can't have as it will allow other countries to ask for the same. And in any case we would be paying for access but lose our seats and say.
Stalemate.
Gibraltar also has a semi-open border with the EU which would remain as it is if we remained in the CU/SM. The problem there is outside the Eu institutions we leave ourselves with the problem of the Spansih closing it completely (they have form in that) and/or refusing those with work permits access (a new problem). Probably won't happen, but may.
Your answer is to give Gibraltar away as it's insignificant.
But by the same token I assume you don't wnat to give the Falklands away, yet arguably Gibraltar is of far more military and geographic significance to Britain than the Falklands, but you want to give it away to suit Brexit??
EDIT the final point - I've just read your new message which kinda answers that!