So my question is how should I handle a player who keeps "suiciding" their character in order to get the stats they want?
2 of the 3 problems you had with this seems to be work-related. The basic problem here being that the player can whip up a new "toon" in an evening, while the job of learning how that character works and integrating it into the ongoing story takes you far more time.
That seems like such a legit concern, that I don't see a huge problem with applying the brakes a bit, on that basis. Tell the player the truth; that adding new characters to the story is far more work for you than merely rolling a new character is for them, and you are going to need at least an extra week to work out the story details. After that, the party has to get themselves to your insertion point, which may well be another week, depending on the choices they make.
Thus any new character will implicitly impose on the player a 1-session "death penalty" where the player will have to sit out. This is one week from when you receive the character info (you can't very well rewrite the story around a character you haven't seen).
If the player wants to work around this by notifying you a week in advance of their old character's impending death, that's actually a good thing. You can instead write it into the story (perhaps the character retires to run an inn), instead of having it end in an unrealistic fourth wall-breaking suicide. Perhaps you can even keep the old character yourself in your stable of recurring NPC's.
As for a new character class that you aren't familiar with, that would be another week on top of the first one so you'll have the free time to familiarize yourself with that class. If the player wants to use a class you are familiar with, they don't have to suffer that extra week. If they really do, then you need that time to learn the new class. Player's choice.
The point is to remove your main pain point in all this: the huge amount of work expected of you on a whim.