What is a bloodrager's caster level before 4th level?

Question

The ranger and paladin classes, core to Pathfinder, are low-grade spellcaster classes that were boosted from their 3.5 incarnations by changing their caster levels to be equal to their class level - 3, in a shift from the prior game's half-level progression. This was generally considered a better-than-prior experience in play at 7th level and above, as the caster level starts to outpace the 3.5 edition rather than lag behind it as the 4th-level paladin or ranger might experience.
But those classes are not the bloodrager, which similarly casts spells (1st-level to 4th-level) starting at 4th class level, but notably lacks the restrictions on caster level or notes about low-level magic item use that the ranger and paladin have. To my best attempt at finding them, anyway.

A bloodrager seemingly has a caster level equal to their class level, but when do they gain it? The Core Rulebook has this to say on the matter:

Caster Level (CL): Caster level represents a creature’s power and ability when casting spells. When a creature casts a spell, it often contains a number of variables, such as range or damage, that are based on the caster’s level. (CRB 11)
A spell’s power often depends on its caster level, which for most spellcasting characters is equal to her class level in the class she’s using to cast the spell. (CRB 208)

What class level is required to, for instance, take the Craft Wondrous Item feat (Prerequisite: Caster level 3rd) as a bloodrager?

Answer 1

The rules on magic say the following about a creature's caster level:

A spell's power often depends on its caster level, which for most spellcasting characters is equal to her class level in the class she's using to cast the spell.

(Emphasis added by me)

Level 4+: Caster level = Bloodrager level

Unlike the paladin or ranger, the bloodrager's spellcasting feature does not say that their caster level is less than their class level. So we take the default rule, where they use their class level.

Level 1-3: Caster level =N/A

Prior to level 4, the bloodrager cannot cast spells.

Beginning at 4th level, a bloodrager gains the ability to cast a small number of arcane spells drawn from the bloodrager spell list.

Until level 4, the bloodrager is not a spellcasting class. Yes, they have a spell list and can use the appropriate wands and scrolls, but there is no indication that they actually have caster levels. This means the bloodrager skips caster level 3, and won't qualify for the Craft Wondrous Item feat until level 4 or higher, when they gain spellcasting.

Jumping from caster level 0 to caster level 4 is a strange progression. It's weird, and the designers have actually addressed this issue on the Paizo FAQ site.