- Battlecruisers have size/power like a battleship, but armor like a cruiser
Not exactly right. Look at German battlecruisers - they have pretty good armor (battleship-grade), but weaker firepower.
The "battlecruiser", in historical therms, is, actually the fast unbalanced capital ship. I.e. capital ship, that sacrifice some characteristics (usually armor, or firepower, or seakeeping ability) to be much faster than cotemporary battleships. The existence of World War I battlecruisers were generally the technical compromise; the 1910s technology was simply unable to combine high speed, battleship firepower & armor in reasonable-sized hull. As soon as more compact and effective powerplants became avaliable in 1930s, battlecruisers disappeared from any shipbuilding proposals, being replaced by fast battleships. Which combined speed, firepower and protection in well-balanced mix.
The "big cruisers" & "small battleships" of 1930-1940s - like French "Dunquerke", German "Sharnhorst", USN's "Alaska", and Soviet unfinished "Kronshtadt" - weren't battlecruisers. They were specific ships, designed because of limits, put on capital ships & cruiser construction by Washington Naval Treaty. Some of them were planned as "Treat-type cruisers killers", some of them - as "supercruisers", some - as "counter-supercruisers".
When the Russians fielded the Kirov class battlecruiser, the US Navy re-activated the WW2-era Iowa Class battleship. The idea was that since the Russians had fielded a huge surface ship again, the US Navy needed something that could survive direct punishment from such a ship, with tons of armor.
Not even close.
The USN reactivated "Iowa"'s, because they need large ships to carry a lot of "Tomahawk" missiles (both land-attack & anti-ship). In early 1980s, "Tomahawks" could be carried only in box-type above-deck armored launchers (called ABL - Armored Box Launcher) -
http://www.thinkdefence.co.uk/wp-content/uploads/2015/01/Tomahawk-Armoured-Box-Launcher.jpg
- and those launchers were bulky and heavy. Most of USN cruisers & destroyers could only carry a pair of such things without being dangerously unstable.
The "Iowa"'s, due to their great size, could carry eight ABL's - i.e. 32 "Tomahawk" missiles. And they were in pretty good conditions. And also the marines really liked their big guns as shore support weapon.
Considering the "Iowa" vs "Kirov"... The "Iowa" have pretty little chances, frankly, even against missile-armed fast attack crafts. No armor could protect against something like "Granit" cruise missile, which dive upon target ship at Mach 3. And active defenses of "Iowa"s, even after refit, were, frankly, pathetic; their CIWS were the only weapon that could actually stop any missile.