Lagrange multipliers is overkill when one variable is a function of the other, as here where $y=4000-x$. Then it's just a matter of maximizing
$$ (3000+x)(9000+4000-x)\frac 13 = $$
The factor of $1/3$ is a constant and irrelevant for the optimization (so it doesn't matter which city is the expensive one, or by how much), and what is left is just the product of two linear expressions whose sum is constant. It is clear even without calculating anything that such a function has its maximum when the two factors are equal -- its graph is a parabola, so its apex is halfway between the roots.
So ideally the mother should split her gift such that the two children's total is as close to each other as possible. The $x=5000$ solution you've already found corresponds to giving $5000$ to child A and $-1000$ to child B which make both children end at 8000 -- but since you can't give negative gifts, the best thing the mother can do is giving all 4000 to A, which will push the product-of-consumptions as high up the parabola as she can.
However, it really doesn't make much sense to add an income (which is per some-unit-of-time) to a one-time gift. So it is tempting to understand the problem differently: The children's current incomes are red herrings, just like the cost-of-living factor already is, and what we're being asked to maximize is the product of the additional consumption each child can make from the gift. Then we're maximizing
$$ x(4000-x)\frac13 $$
and just as before the $\frac13$ factor is irrelevant, so the way to maximize the product is to give each child the same dollar amount, that is, \$2000.
Which kind of parent gets the crazy idea to maximize the product of their children's purchase power, though? It does lead to seeking equality between the children, which is a noble and fair goal, but why on earth phrase it as maximizing a product?