First, I'll say that I agree with Icyfire and start from there, but I feel most of the answers aren't answering the question as framed. My suggestions for that are on the second section.
Even if you do share the spellbook, it's not that bad
As a Wizard player, if my DM just told me "You have access to every Wizard spell" (just like Clerics and Druids do), it wouldn't change too much. Yes, Wizard spell-list is more versatile and broad, but the main limitation is still number of prepared spells and only being able to cast one concentration spell at a time. Also, unless you allow very creative ways of casting spells, honestly there is a considerable difference between the top-tier spells (which are usually 4 for each spell level, max 6) and the others, so 4 spells is usually enough to get the spells you are actually going to be using most of the time.
If you just give the spellbook, it changes even less - the Wizard still has to use a lot of gold and time to copy the spells, and as you mentioned, the cost is non-trivial.
As a note, the main reason (I think) Wizards don't know every spell from start like clerics and druids do and the main thing you would break if you gave him every spell in the game:
Wizards can cast non-prepared rituals, so take care mainly with the Ritual spells you are going to put on the book.
But if you really don't want to, there are a few reasons to not lend it
- I will start with the main one, from my point of view as a Wizard player: A wizard's spellbook is its most valuable item (unless he also has a Staff of Magi, then that's that), and I'm not talking about gold. It's all his research, his own annotations, it's everything he worked as a class-worker since ever. "He trusts the party, maybe not completely" - honestly, my Wizard would probably lend a copy of his spellbook to his parents. Unless this trust is incredibly high, it's not enough to lend your most personal treasure.
- Without more details on your setting, the following suggestions might be a blind shot, but... as I mentioned, copying the spells takes time. During this time, the NPC is without his spellbook, meaning he is limited to the spells he has prepared. He might fear being attacked and not being prepared during this time.
- He might trust the party but think they are too weak (although he's only 1 level higher) to protect the spellbook. This actually leads to my third "big" suggestion.
Note: The last two points and the following assume they weren't going to stay on NPC-Wiz house resting while copying the spellbook. You can make some excuses to why they can't if needed, or can force these solutions even if they do.
You can share the spellbook and then have some thieves steal it from the party
And there you have your sidequest (recovering the spellbook, so they don't lose NPC-Wiz trust and actually get the spellbook), in which the reward is the spellbook, without making your NPC-Wizard being the one requesting things from your party to lend some help and also without just giving them more power without any effort or consequence.