This is a bad idea... allow me to explain why.

In 1.0, SE allowed people to change classes mid-dungeon as you are suggesting. This resulted in a lot of class stacking for certain bosses. One example is Aurum Vale. The first boss, all of the dps went BLM. The final boss, all of the dps went MNK. Class stacking FORCED people to have multiple jobs at 50 and the ability to switch. SE wanted ARR to be more friendly toward people who only had one class geared -- so what you enter a dungeon with is what you are stuck with.

Regarding your mention that it should 'only' apply to like-minded classes (SMN/BLM, WHM/SCH, PLD/WAR, etc), this would still not be a good idea because each class has its strengths and weaknesses. I'll talk about SMN/BLM because those are two classes I know well. SMN is better on single-target when you have to dodge a lot of things. BLM is better on adds you have to burst down really fast. These are two distinct strengths/weaknesses that force you to form a balanced party. If you could switch mid-dungeon, it would remove the requirement of entering with a balanced set of people. Everyone would switch to SMN for dodging fights, or BLM for 'bursting down the adds' fights. Of course, there are a bunch of differences between SMN/BLM but I'm oversimplifying it to keep this post short.