The solution is this: each card should have two abilities. One should be useful on its own and one should augment the next piece in a way that doesn't affect the third piece. Specifically, two abilities need to change.
1. The ability on Spex Prism, "Cards named Spex Lighthouse have "

: Untap Spex Mirror." " needs to change into something that augments Spex Lighthouse in a way that
doesn't involve Spex Mirror.
2. The ability on Spex Lighthouse, "Whenever a card named Spex Prism comes into play, put a charge counter on it." needs to change into something that is useful on its own. Spex Prism already comes into play with a charge counter on it and Spex Lighthouse already has an ability that augments Spex Mirror.
May I suggest the following:
Spex Prism

Artifact
Spex Prism comes into play with a charge counter on it.
Remove a charge counter from Spex Prism: Add one mana of any color to your mana pool.
Cards named Spex Lighthouse have "

: Untap target artifact."
Spex Mirror

Artifact

: Add

to your mana pool.
Cards named Spex Prism have "

: Put a charge counter on Spex Prism."
Spex Lighthouse

Artifact

,

: Search your library for an artifact card and reveal that card. Shuffle Spex Lighthouse into your library, then put the card on top of it.
Cards named Spex Mirror have "

: Add

to your mana pool."