Can an Artificer with an All-Purpose Tool gain Eldritch Blast cantrip and take the feat Eldritch Adept to get Agonizing Blast Invocation

Question

Can an Artificer with an All-Purpose Tool gain Eldritch Blast cantrip and take the feat Eldritch Adept to get Agonizing Blast Invocation?

All-Purpose Tool: [...] As an action, you can focus on the tool to channel your creative forces. Choose a cantrip that you don't know from any class list. For 8 hours, you can cast that cantrip, and it counts as an artificer cantrip for you. Once this property is used, it can't be used again until the next dawn.

Agonizing Blast - Prerequisite: Eldritch Blast cantrip - When you cast Eldritch Blast, add your Charisma modifier to the damage it deals on a hit.

If the Artificer chooses a different cantrip the following day, they still have the invocation right, it just doesn't function till the Artificer chooses Eldritch Blast the day after?

Answer 1

Eldritch Adept does not give access to invocations with prerequisites unless you are a warlock.

Eldritch Adept states:

If the invocation has a prerequisite of any kind, you can choose that invocation only if you’re a warlock who meets the prerequisite.

Since Agonizing Blast has a a prerequisite, you cannot take it unless you are a warlock, even if you know or can cast eldritch blast.

Answer 2

Even if Eldritch Adept allowed it, you wouldn't fill the prerequisites.

The Agonizing Blast invocation requires you to know the eldritch blast cantrip - the All-Purpose-Tool gives you the ability to cast it, but that is not the same as knowing it - otherwise, for example, holding a wand of magic missiles would let you also cast the spell using your spell slots.