a damage buff will increase the over all attack your assaulter's have so say in your partner tab you have 10k damage that 10k damage will be increased by 8% so you will have 10,800 damage instead. Damage rate works a bit different. Damage rate is applied after the game calculates the difference between your attack and the enemy's defense then it applies the damage % increase to the value that remains. This is what i've noticed anyway. So you can use a zan like zabimaru and nano.
If its like soygo and nano though the damage buff will go with which ever one is stronger.