You answered it yourself. NO.
As you said, by RAW, casting Find Familiar will get rid of any previously existing Familiar and that is the end of that. It does not say it has to be from any particular class.
As far as "should I allow them to do it anyways?" I would still suggest no. Battles are only going to slow down with having extra creatures and one of the players is going to have triple the amount of things to do compared to the rest. This is without even considering any possible exploits that could come from this.
Making the familiar stronger also has it's own cons, as a "pet" that is comparable to other PC characters can seriously shift the balance of the game, especially for that character who only needed to be level 4 to play a "second character". If you take this route, it should not be something like an owlbear but another small utility outsider like the ones Warlock gets.
It would be a more compassionate case if not for the fact that Wizard can just pick different spells and maintain their full effectiveness. He's not "losing" anything with this multiclass.
If you still want to give him something, let him change his pact to something else or add some new options to the familiar list that aren't stronger than the unique ones he already gets.
No
From a lore perspective, definitely not. To begin, Wizards and Warlocks use different methods of producing magical effects. Wizards study for years to understand how the Weave works. They experiment with and manipulate arcane powers to develop spells.
"They learn new spells as they experiment and grow in experience" PHB 112
Warlocks, on the other hand, channel the power of their otherworldly patrons and do not understand the inner workings of that magic themselves.
"The magic bestowed on a warlock ranges from minor but lasting alterations to the warlock's being... to access to powerful spells." PHB 105
"Various patrons give their warlocks access to different powers and invocations..." PHB 108
What this means is that your warlock doesn't know enough about the spell to actually teach your wizard how to cast it. This would be like asking for directions to bake a cake but only being told to heat eggs, flour, and milk in the oven.
From a technical perspective, also no. Wizards can learn new spells from various sources, including other people. However, the PHB is very specific about those people also being wizards:
"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. PHB 114 (emphasis added by me).
"They can also learn them from other wizards, from ancient tomes or inscriptions, and from ancient creatures... that are steeped in magic." PHB 112 (emphasis added by me).
Now, nowhere in the PHB is it explicitly stated that a wizard cannot learn spells from a warlock, but it is very heavily implied.
On a side note and as a friendly reminder, do keep in mind the purpose of RPGs. You're coming together as a group to create a story. The official rules are guidelines for that story, but the players and the DM are the ones writing it. If you think that learning a spell from the warlock would make the story better by providing deeper interaction and character development between your character and the warlock or enhancing the current plot, then consult your DM about that between sessions.
Best Answer
No
Your plan will not work for 2 reasons:
Bards cannot teach wizards spells
The Player's Handbook outlines the ways that a Wizard can learn a new spell via Wizard class features:
By leveling up (research)
Now, thematically, you might say that, as part of the Wizard's research, the Bard was able to help them learn and master the spell. However, this would not grant you any additional spells beyond what the Wizard would normally be able to learn on the occasion of their leveling up. So this would be a flavor-only thing and not mechanically helpful which it seems is what you want.
By finding spells on scrolls or in spellbooks and copying them into their spellbook
In general, the way a Wizard learns spells is by deciphering some written form of the spell and then laboriously copying it into his own spell book.
Note how this quote emphasises that the material they learn from is generally assumed to be 1) written and 2) written by a wizard specifically.
So there is no mechanical way for a Bard to teach a Wizard a spell.
Not only does it not work mechanically, it also doesn't make sense thematically since Bards and Wizards have very different ways of casting and their techniques are just not transferable.
It would not help with transcription costs
"We're trying to allay costs for transcribing spells a bit"
This part deserves some attention as well. Aside from leveling, using any other method in the Wizard class to add spells to your spellbook require you to take the time and pay the cost associated with adding that spell to your spellbook.
So even if a Bard were to teach the spell, you would still have to transcribe it into the Wizard's spellbook and thus take the time and costs specified in the book.
Caveat
As always, the DM may allow this to happen in some form, but just know that that decision is neither supported by the rules nor the flavor of the Wizard and does not change what the rules say about it.