the paladin is a lvl 2 War mage he can use lvl 1 and 2 war mage only spells... but not lvl 3+ war mage onlys
^This is exactly correct. The Paladin only counts as a War Mage for the purpose of level 1 and level 2 War spells.
This includes spells that combine into a higher level than 2 but only have 1 or 2 levels of the War school. Even though Bridge Troll is Nature 2 & War 1 (and thus a level 3 spell) the paladin pays 5 spell book points for it, 4 for nature and 1 for War.
A level 3 War spell such as Dwarf Panzerguard would cost the Paladin 6 spell book points to include in his book because he does not count as trained in level 3 War spells. Similarly the Paladin may NOT use Akiro's Hammer but is allowed to use Ballista.