Two different SPI modes in one microcontroller
To try and answer all aspects of your question I quickly pulled up the relevant pages of the data-sheet. Unfortunately the Atmel download doesn't work right now, so I'm using the old one from my archive that only goes up to mega168(P)(A) devices, but they probably haven't removed any functionality. I'm even doubtful they changed anything other than the memory specs.
So, I'm not going to link any pages, graphs or tables, because that would be confusing without a direct link to the one PDF I'm looking at.

One aspect of my answer is: Yes, you can switch between modes if you are not actively clocking to any of the connected devices, which means keeping their Chip-Selects unasserted is the easiest way of making sure. Then when you are in the one mode, make sure you don't accidentally assert the wrong chip, because you might be making choices on faulty data. If you have that secure you should be golden. You would be well of to switch over and have about 1.5 bus-clock cycle of waiting (interrupt based or otherwise), as the internals of the SPI device may possibly show some transient behaviour.



Edit1: Based on the post by Markt and commenting there, the turn-off, change, turn-on procedure may well be faster than waiting 1.5cycles of a 1MHz bus, as the core running at 10MHz+ will probably handle these instructions in less time than that. And even if that takes a tiny bit longer, it will be more reliable and easier to guarantee (wait 1.5 cycles isn't the most trivial thing right out of the box).



However, are you aware that the USART can also operate in Master SPI mode 0, 1, 2 and 3? In my data-sheet after the section about USART0 it shows a section specifically called "USART in SPI mode".
If you are very worried about difficulties switching over, and possibly missed signals (I think missing signals is very doubtful if you plan your switches out well enough, but still) or just about total throughput with hosting both devices: Why not use the USART as another SPI port?