Then it's not 2x claws and 2x slams, it's 2x claws and 1 slam then. The damage numbers are then lower (2d4+4 and 1d8+4 equals to 17.5 damage on average, while (2d6+6)*2=26 damage on average, if it hits. Of course, the chances to hit are lower. It depends on enemy's AC how lower they actually are

1) d20pfsrd.com/magic/all-spells/l/lead-blades A Ranger spell that costs 15 gp per use from a wand.

It increases weapon damage by one size category and stacks with Enlarge Person.

It's Range: Personal normally, but with a good UMD score one can use it on their Eidolon as Range: Close

2) Finally, an Azata can gain an Exotic Weapon Proficiency for just 1500 gp. d20pfsrd.com/magic-items/wondrous-items/h-l/ioun-stones/…

3) Once the prerequisites are made (dex 21), d20pfsrd.com/feats/monster-feats/multiweapon-specialist Multiweapon specialist also becomes a great choice, giving a flat +2 damage for each damage roll

Not sure if it's actually worth a feat, as it pretty much requires a Dex build