The 30-06 bolt face will need to be opened to fit anything bigger than the 30-06 family. In that case, I would keep the '06 or go to 280 Remington.
PTG will make a bolt to fit which ever caliber you decide. I would go with a 300 Win due to the balance between energy and barrel life. The 300 RUM is hard hitting but barrel life is limited. The 6.5-06 is real flat and bullet selection is awesome but again, your barrel will suffer. My last 300 Win barrel saw 1147 rounds before it had to be replaced. Some get more, others less but 1200-1700 is expected.
I wouldn't say the 6.5-06 is a barrel burner. It is a pretty mild 6.5mm cartridge compared to the larger 6.5s such as the .264 WIN Mag and 6.5 WSM.