Jason, yes, on some bikes the trans will drain while upside down. Some will get air in the fork and shock oil too, but that isn't a concern if the bike is inverted for a short time only.
I forgot to say that getting some raw two stroke oil in the bottom end right away is a good thing too(same thing, you can feed it directly there by going up the exhaust port or plug hole at bdc and dripping oil down one of the transfer ports-crazy straws work well down the plug hole).
Then plug out and a bunch of kicks or push it in gear just to make sure the bearings are all coated well and any coolant mix left is very dilute.
If bq does this and still sees coolant then something else is wrong, he has a coolant leak somewhere(base gasket is rare-not many have coolant passages, more likely a head gasket issue).
Ohh yeah, when it does fire, do all you can to make sure it stays running until hot. It will prolly miss a bit and smoke some, and you may need to remove the plug and dry it if it dies, but do that until it is hot hot. Then a new plug and go ride.