Within the narrative, you don't have stat blocks in your spell
Other answers here have capably described the rules regarding how the Awakened Spellbook works with regards to damage type. I think it is also worth at least considering how this would function on a narrative level. We know that the spells in spellbooks are complicated lists of formulae, gestures, and sounds of power:
From 'your spellbook' sidebar in the PHB:
Copying that spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.
Within the narrative, we can understand how the Awakened Spellbook could take an arcane symbol representing fire damage in one spell and draw on it to replace the rune representing the force of cold in another spell so as to alter a spell when cast.
We can understand how a spellbook might contain the ritual words and acts needed to summon and bind a fey creature to one's service.
But even within a playstyle where the characters in-game are aware of many game mechanics as such, it seems incongruous to expect the arcane formulae of the spell to come with a section saying 'here is the statistics of the fey you are summoning'. While I can imagine a wizard annotating their spellbook with a description of the effects of a spell, in Common, and even noting that the fey summoned typically come with arrows, such an addition would not be contained within the spell itself, and thus would not fit the condition of (emphasis mine):
you can temporarily replace its damage type with a type that appears in another spell in your spellbook".
It is difficult to see how such non-arcane marginalia in a spellbook could be used by the Awakened Spellbook to allow another spell to do piercing damage.