It seems that every 4 or 5 months, this subject [or very similar] does it's rounds. I'm not going to state how many moose I have shot for fear of being called a liar. Suffice to say it is plenty.
As many have stated, moose are not hard to kill [Elk are considerably harder to put down quickly] I have shot moose with the following chamberings: 6mm Remington, 257 Roberts, 6.5x54 MS, 6,5x55 Swede, 264 Win Mag, 270 Winchester, 7x57, [8 moose with this one] 7mm Rem Mag, 7mm STW, 300 Savage, 308 Win, 30-06, 300 H&H mag, 300 Win Mag, 308 Norma Mag, 303 British, 30 Gibbs & 8x57.
[While I would hesitate to call the 6mm and the 257 R moose chamberings, they will do the job with a good bullet placed properly.]
I have only once had a scare from a moose that was shot and had fallen, but he only managed to get to his feet...the life was gone from him, and he collapsed quickly again. The 270 is plenty of gun to kill any moose that ever walked, and I wouldn't hesitate even of he was inside of 20 yards.
I do not believe any chambering listed above stood out as being a "quicker" killer on the big cervids. The vast majority of the moose I have shot have been one shot kills. Use your 270 with confidence! Regards, Eagleye.