I would give it a shot... Just remove the inflated battery, plug in the power brick. and try to fire it up. The worst that will happen is that it will not work, and you will still have to wait on the new battery. Don't leave the inflated battery pack in your machine... Blown lithium packs can auto-ignite if they continue to be charged by the power brick.
My Alienware 18 worked fine with a completely dead battery while I waited for my new battery. - Though, I could leave the old one in the machine because it was not inflated like yours. - It just would not charge.