Ultimately, you as DM decide what you believe is plausible
If you decide that it's unreasonable for a PC to "quickly find a rare animal in hundreds of square miles of forest", then you are well within your rights to not let that happen. Explaining this to the player up front was the right thing to do, since it sets his expectations immediately rather than potentially drawing it out.
Furthermore, although players can ask for things they want, you are under no obligation to do that. If I was a player in your game and I demanded a certain magic item, I should not expect you to oblige me, nor if I wanted to recruit allies for free to accompany the party (since that makes the party stronger and more likely to survive, which affects the balance and therefore your preparation as DM; it's different if they buy hirelings or something, since at least that's using a game resource, money, but just expecting to "find a pseudodragon because I want one" isn't good enough).
Unless the rest of the party are all really genuinely on board with the plan of getting this one player a pseudodragon, then it would be unreasonable to expect them to have to go along with this player's plan. It would also be unreasonable to have to split the party, since, ultimately, D&D is a game about team work, with multiple players collaboratively playing together, and the players are expected to generally stick together, not go off on their own for weeks at a time whilst the rest of the party continue with the actual story.
Potential balance issues with having a pseudodragon
This pseudodragon, if you were to give him one, would effectively be an ally, and given that I imagine you'd have to use the Variant Familiar version from the Monster Manual (p. 254), would also give him a significant buff (specifically that the pseudodragon can share its Magic Resistance trait with him, so he would now have advantage on saving throws against spells, effectively "for free") that presumably the other players would then be missing out on, which may make them all feel left out.
That could then snowball if you feel the need to give them all something to make up for it, since now everyone's power level increases, and that unbalances the game (in the sense that you, as DM, now have to take that into consideration when balancing combat encounters, etc).
There is a RAW-based solution that doesn't require you to go back on yourself
Furthermore, there is a way that the player can just "have" a pseudodragon without making you give him something "for free", and that's taking 3 levels in warlock and going Pact of the Chain, which allows his find familiar spell to summon a pseudodragon. If his plan was to eventually learn true polymorph and turn into a dragon, that would still be possible by level 20 (and no sooner, since he'd need to be Wizard 17/Warlock 3 to have both true polymorph and the warlock familiars). This is all assuming you allow multiclassing, of course...
What about potentially losing a player?
If the player cannot handle not having something that they have no right to expect in the first place, then it speaks poorly of this player's maturity. If he leaves because of that, then I don't think anyone can blame that on you. As for how this may affect the other players, it might be best to discuss that with them at the time to see where they see the game going after that, if this player's loss would bother them.
However, let's not be too pessimistic. Although you suspect that this player may not be able to handle it, this may just be guilt or a lack of confidence in your DMing speaking for you. This player may handle it better than you expect. It's possible that this player will accept that he didn't get what he wanted and move on, and that there will be no further effects of your ruling. If you have confidence in your rulings as DM and believe you are doing the right thing, taking into account balancing the players' fun (as a whole group, which includes you as well) with what's plausible, then it may be that no problem occurs at all.