The Large strix is wielding two inappropriately sized weapons, a bastard sword designed for a Medium creature in his main hand, and a longsword designed for a Medium creature in his off-hand.
While a Large creature wielding a weapon designed for a Medium creature reduces the effort needed to wield such a weapon by one step (from two-handed to one-handed and from one-handed to light), the Large creature still suffers the penalty for wielding a weapon designed for a creature not of his own size category according to the section Weapons on Weapon Size under Inappropriately Sized Weapons.
This strix suffers a –2 penalty on attack rolls when making standard attacks with these weapons and when making a full attack with one weapon or the other. But when two-weapon fighting using both weapons the strix uses the chart below, which I've modified and extended owing to your circumstances from the section Combat on Two-weapon Fighting.
Table: Two-Weapon Fighting Penalties (Extended)
| Main | Off- |
Circumstances | Hand | Hand |
-----------------------------------|-------|-------|
Normal penalties | –6 | –10 |
-----------------------------------|-------|-------|
& main weapon 1 size wrong | –8 | –10 |
& off-hand weapon 1 size wrong | –6 | –12 |
& both weapons 1 size wrong | –8 | –12 |
-----------------------------------|-------|-------|
& off-hand weapon is light | –4 | –8 |
& main weapon 1 size wrong | –6 | –8 |
& off-hand weapon 1 size wrong | –4 | –10 |
& both weapons 1 size wrong | –6 | –10 |
-----------------------------------|-------|-------|
With the feat Two-weapon Fighting | –4 | –4 |
-----------------------------------|-------|-------|
& main weapon 1 size wrong | –6 | –4 |
& off-hand weapon 1 size wrong | –4 | –6 |
& both weapons 1 size wrong | –6 | –6 |
-----------------------------------|-------|-------|
& off-hand weapon is light | –2 | –2 | <- This is where you want to be.
& main weapon 1 size wrong | –4 | –2 |
& off-hand weapon 1 size wrong | –2 | –4 |
& both weapons 1 size wrong | –4 | –4 | <- This is where you are.
-----------------------------------|-------|-------|
The strix's current bastard sword and current longsword deal 1d10 and 1d8 points of damage, respectively. Were the strix to employ a longsword designed for a Large creature and a short sword designed for a Large creature, the strix would no longer be using inappropriately sized weapons and with this new longsword and short sword would deal 1d10 and 1d8 points of damage, respectively.
Yes, that's exactly the same amount of damage. (For reals!) In other words, this strix would be far better off with weapons appropriate to his size. The only thing good about these littler weapons is that they're probably easier to find.
The feat Lighten Weapon only helps with bigger weapons
The third-party feat Lighten Weapon says
Choose one type of weapon. When using a weapon of that type you may reduce the effort required to wield it by one step but suffer a −2 penalty to hit with that weapon. This allows you to wield a weapon 1 size category larger as if it were your own size, use a two-handed weapon in one hand, or a one-handed as a light weapons.
As evidenced by the feat's example, here type means a specific weapon (in the same way one must choose a weapon for the feat Weapon Focus, for example), so a Large strix possessing the feat Exotic Weapon Proficiency (bastard sword) who takes the feat Lighten Weapon (bastard sword) could use a bastard sword designed for a Huge creature (that deals 2d8 points of damage), but that weapon would also be inappropriate by 1 size category and, as per the feat, the strix would take an additional −2 penalty on attack rolls with such a weapon.
The trait Giant-blooded should probably only help with bigger weapons
The campaign trait Giant-blooded (Giantslayer Player's Guide 6) says that
When you wield a weapon that is larger than your size, the penalty on attack rolls for using inappropriately sized weapons is reduced by half.
One could argue that while a creature wields a weapon that's inappropriately sized because it's bigger, the creature could simultaneously wield another weapon that's inappropriately sized because it's smaller, gaining the trait's benefit with both. That's pretty shady, though.