I think as long as you are ready to recognize that there may be some differences between the brands or sizes, there is no reason not to mix and match. I have run several different brands and sizes over the years with mismatched sets with no problems.
If you look at racers running sticker tires or scuffed tires or in NASCAR coming in for two outside tires only, they seem to survive.
Again, if there are significant differences in grip, your car will behave quite differently as Jad pointed out.
At the last Willow event, I started with a set of Falkens 225/45/17 front and 245/45/17 rear. Switched the fronts to a set of 245/45/16 used Michelin Sport Cups for several sessions, then put the Falkens back on the front and moved the Michelin to the rear and finally for timed runs went back tot he Falkens again.
Greg