Not possible. Fast type change puts a different chart type over the straight table equivalent of the basic chart. The structure of this straight table equivalent won't change.
As you already mentioned yourself, you could define multiple chart objects and put them in a container, or you can define a few tightly aligned buttons that each show/hide different charts that cover the same area.
That would be the cleanest feature to help solve this kind of "issue".
OTOH, I think it's not that difficult to enable the caption of say 3 aligned graphs, and simulate the Fast type change icon using three iconised graph representations. Those can be put on top of the chart caption and are hidden together with all graphs except the next one in sequence. A bit more work, but looks & operates the same.
The only thing for which I have no immediate alternative is the right mouseclick-menu of the Fast type change icon...