If you are going to move a betta from an appropriately heated tank (78 degrees) to a too-cold tank (70 degrees) I suggest you bag and float for a LONG time to prevent temperature shock. Honestly, I dont' see the point in moving the animal from ideal conditions to a cold, undersized bowl anyways? If it is to treat for an ailment, know that most bettas do better being treated in warmer temps (unless it is columnaris) as thier immune system is more active at higher temps.
I've always heard that the ideal range for a betta tank is 76-80, and anything much below 72 was bad. Not sure about that though. Everyone is right about the tank size though; in addition to being inadequate, it is VERY dangerous to heat such a tiny tank. Could you just float the little tank in your heated 10 to keep the temp up in it? Just a thought, not sure how viable it is.