How often would a Barbarian be fighting but not raging?
Yes. It's best to carry around two weapons. You should always be versatile with the kind of damage you deal. It's also recommended to carry an extra weapon to deal ranged damage, since it's not every fight you'll be able to fight in perfect conditions.
Yeah, any martial character should carry multiple weapons anyways, for damage type, special traits, CritSpec effect, range.
There is Titan Mauler specific rationale to do so, but it's not quite the one the OP imagined... This is what he missed:
When wielding such a weapon in combat, increase your additional damage from Rage from 2 to 6, but you have the clumsy 1 condition (page 618)
So BOTH the bonus Rage damage AND the Clumsy condition only apply in Rage WHEN WIELDING OVERSIZED WEAPONS.
So you don't need to NOT Rage to avoid Clumsy, you just need to use NORMAL size weapons while in Rage.That doesn't get your bonus Instinct damage, but you still get normal Rage damage and Temp HPs and can use Rage Feats.
(most Instincts bonus damage won't always fully apply, i.e. when it is type Resisted by target, so no different here)
In terms of the specific effects of Clumsy...
AC is probably main concern for melee PC, but instead of switching to normal size weapons, consider using Large 1H weapon with Shield. That yields better AC than most non-Shield Barbarians (+Shield Block resistance potentially). Not to say using normal sized weapons isn't also legit option (including WITH a Shield), just this is another one.
Ranged Attack rolls is probably the one you always will prefer normal size weapons, because it is directly penalizing the weapon attack. So although the Instinct isn't restricted to melee weapons, oversized Bows probably aren't a good idea. Of course, melee thrown weapons are grey area in that you might have oversized ones for melee, and that's just what you end up having to use to Throw... especially if it has powerful Runes which make it desirable to use despite penalty. But generally trying to keep ranged attacks normal sized is good bet.
How often would a Barbarian be fighting but not raging?
An extended/multi-phase encounter could put you in this situation, rage lasts 10 rounds and it has another 10 round cooldown.
So although the Instinct isn't restricted to melee weapons, oversized Bows probably aren't a good idea.
The Instinct just adjusts your extra Rage damage, which itself is already restricted to melee weapons and unarmed attacks. And Raging Thrower doesn't extend to bows, so oversize bows are definitely a poor idea.