Yes, it will work.
I did the same thing four years ago with a 12,000 btu portable and a deep cycle battery.
My application was a little different. I used the a/c at night in a 28' Rinker cabin cruiser on a trip down the Arkansas River to the Mississippi.
I did a test before the actual cruise. I took a fully charged battery, hooked up the inverter and let it run in the garage. I lasted four hours before the inverter shut off because of lack of power in the battery. Since I was using it at night, the four hours was enough for me.
I recharged the battery each day as I continued on my trip down the river.
Just do a few tests and see what happens.
You may need two batteries, depending on how long you want to run your a/c.
Let us know how your test works out.