It's almost correct, but not quite.
There are two separate, unrelated ways to get a familiar in D&D 5e, and you're trying to combine them.
The first method is by use of the Find Familiar spell, which allows you to summon a celestial, fey, or fiendish spirit that takes the form of any of a list of creatures. This list is expanded by the warlock pact of the chain. This familiar is perfectly obedient, can be resummoned when it dies, can be hid in a pocket dimension, deliver touch spells that you cast, and everything else specified in the spell description. This method gives you a familiar with the basic stats of the chosen creature, not the "variant: familiar" traits of the chosen creature (unless your DM chooses to have that creature type appear).
The second method is by finding a quasit, imp, or pseudodragon that has the "variant: familiar" trait (which is 100% up to the DM), and enlisting it as a familiar by interacting with it. This familiar has only the traits listed in the stat block for that creature, including the variant traits, but none of the traits of familiars given by the Find Familiar spell. No pocket dimension, no touch spells, no limitations on what actions it can perform, and if it dies, it's just dead. This type of familiar is an NPC controlled by the DM (much like a hireling or other follower), and is only as obedient to the PC as the DM says it is, using the MM entry as a guide.
Now that we've established how things actually work, we can address your real concern. Your warlock can't yet communicate at a great distance, but he can soon. There is a warlock invocation available to him called Voice of the Chain Master that does the same thing, but with unlimited range on the same plane. As you've realized, this ability has some incredible potential, especially for scouting.
It's not any more powerful than other options and should not be limited.
Let's compare it to some other invocations. There's one that lets a warlock cast Disguise Self as at will. This would let him see a guard, and appear exactly like that guard and just walk around the enemy camp unimpeded. Or maybe impersonate the leader of the camp and just take it over without even a struggle. There's another invocation that lets the warlock cast arcane eye at will, which gives you a way better scout than an easily killed creature. A familiar, even an invisible one, still has to succeed on a Dexterity (stealth) check to avoid being heard and then easily killed. An arcane eye does not.
So, in order to fully utilize this scouting ability, your warlock has to pick the chain pact and spend one of his few, precious invocations, both of which are huge opportunity costs. He deserves something in return. This something is you not limiting it. It already has a flaw in still being able to be killed by anything that hears it, or smells it. That's right, just about any pet wolf is going to be enough to catch this familiar. It doesn't need any more limitations.
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.
Best Answer
The closest suggestions to what you want that I have found are a Beast Master ranger or the find [greater] steed spells
I was unable to find anything that allows a wizard/warlock to have self-only spells also affect a familiar summoned via find familiar. The closest things I could find would require heavy multiclassing into undesired classes, and still wouldn't actually affect the familiar, but here's what I found...
The Beast Master ranger gets a class feature at 15th level called Share Spells (PHB, p. 93), which says:
This doesn't affect a familiar via find familiar like you wanted, but it does allow your beast companion to benefit from Tenser's transformation. Note that with 15 levels in ranger, you wouldn't have access to Tenser's transformation anyway.
Alternatively, if you were able to gain access to the find steed (PHB, p. 240) or find greater steed (XGtE, p. 156) spells (which, as far as I'm aware, means taking levels in paladin or bard1), both of which say:
Again, this still doesn't affect the familiar via find familiar, but this should also allow a summoned mount to be affected by Tenser's transformation. If you took the bard route, you could learn Tenser's transformation via Magical Secrets, otherwise this build also doesn't have access to Tenser's transformation.
1 Actually, there is another way for this to work; rather than being of a class that can cast either find steed or find greater steed, you can instead use the Ring of Spell Storing, which can hold spells of 5th level or lower. If you know a paladin or bard with either of the steed spells, they can cast it into the ring, allowing you, a wizard/warlock, to cast find steed or find greater steed, and thus still having access to Tenser's transformation via wizard levels. Of course, none of this resolves the problem that it doesn't help with your familiar via find familiar...