Very interesting thread, have to admit! There's lots of pieces to this puzzle and only so little pieces of it are in the public domain.
I've been trying to find good quality images of the #61 of the last JGTC'96 round (Mine) but so far not so succesfully. Those would at least give some hint which chassis it could be. The video clips on YouTube of the season round up
part 2 are such low quality that the details to see on the #61 car aren't much help on more specific details. But if you really look from 4:21 onwards, it seems that the #61 car has been upgraded to the GTR'96 specs (especially the front air intakes) and as we know 08R (which many has guessed was the car replacing 14R) was never upgraded to GTR'96 specs as can be seen from the recent pictures from Bahrain (or why would have they then converted it back to GTR'95 specs?).
Also one piece to that is that some while ago in some interview of a McLaren spokesperson of the customer care or similar (can't really now remember so specifically) said that to his records none of the GTRs is written off, all of them repaired (if needed). If true, which I also believe (24R was repaired after almost full destruction in Suzuka'97!), 04R is repaired and still somewhere in existense, but hidden away. The time frames would allow it well to be the Mine #61 upgraded to GTR'96 specs and with a new engine/gearbox/etc. And maybe someone then after the last race bought/transferred it to his/her/their collection as the real JGTC'96 championship winner in the famous Lark livery. Just one possibility.
So which chassis are the other possibles to have replaced the 14R in such short time? I think the real possibilities are quite limited. We could try to start this puzzle by outruling the impossibilities first, collecting as much evidence as there is about the Mine #61 racer, the time frames of different chassis around that time (their whereabouts), and so on.
And a good start would be if LOJ himself could give some info did he sell the damaged 04R back to McLaren as he bought the 08R, which would have been a rational choice at that time.