The Blood Arcanist loses her Arcanist Exploits at the 1st, 3rd, 9th, and 15th level. My question is if this also means that the Blood Arcanist does not count as a character that possesses the Arcanist Exploit class feature until 5th level. I ask because I was considering taking Extra Arcanist Exploit at 1st and 3rd level, but I wasn't sure if doing so would be legal.
I don't think that's legal by the RAW. The rules on archetypes specifically say that:
A character who takes an alternate class feature does not count as having the class feature that was replaced for the purposes of meeting any requirements or prerequisites.
The wording of the blood arcanist ability states that:
This ability replaces the arcanist exploits gained at 1st, 3rd, 9th, and 15th levels
At 1st-4th level you have not been granted the arcanist exploit class feature, you don't qualify for the feat.
So RAW I don't see much wriggle room. But since the Blood Arcanist does eventually get exploits, and they function off your full, normal arcanist level, I don't think there'd be any balance reason to enforce this rule.