This is a tough one. I think I would prefer option 2 out of those listed in your first post. I think Jaime's loyalty at this point was probably mostly geared towards Cersei (and Joffrey indirectly). However, I just can't picture him as a general in Joffrey's faction because he was commanding a Lannister army at the time this mod starts. I'd be fine with him being heir to Casterly Rock because I think his whole outlook, his loyalty towards the Kingsguard and loyalty to Cersei changed drastically after his captivity and losing his hand.
As far as Tyrion, an argument could also be made that he belongs in House Lannister since he did return to fight for his father after he left the Vale. I think the decision was made to include him in Joffrey's faction because he served a more significant role in protecting King's Landing and during his time as the Hand. Moving him to Lannister would also leave Joffrey a little short handed in terms of generals.
Tough decision which I think warrants more discussion.