I'm not much good at even basic number theory, so this is really mysterious to me. I think the actual question might want to restrict to finite Laurent series, but I don't think it matters: I'm pretty sure what I want to do is figure out what each homogeneous component is allowed to look like. Moreover, I'm pretty sure that by plugging in $h=k=1$ I get that all the coefficients in nonnegative degrees need to be integral, so the first place to look is in degree $-1$.
So suppose we have \begin{equation*} f(u,v)_{-1}=\frac{a_1}{a_2 u}+\frac{b_1}{b_2 v}. \end{equation*} Then \begin{equation*}f(ht,kt)_{-1}=\frac{a_1}{a_2 ht}+\frac{b_1}{b_2 kt} = \frac{a_1 b_2 k+b_1 a_2 h}{a_2b_2hkt};\end{equation*} for this to lie in $\mathbb{Z}[t^{\pm},1/h,1/k]$ we need $a_2b_2|a_1b_2k+b_1a_2 h$. I believe it's a basic fact that since this has to be true for all nonzero $h$ and $k$, we need that $a_2b_2|gcd(a_1b_2,b_1a_2)$. This implies that $a_2b_2|a_1b_2 \Rightarrow a_2|a_1$ and similarly $b_2|b_1$. But assuming we started with our fractions in reduced form, this implies that $a_2=b_2=1$; that is, $f(u,v)_{-1}\in \mathbb{Z}[t^{\pm}]$.
Since I don't think I gained much insight from that calculation, I just tried to find restrictions on \begin{equation*} f(u,v)_{-2}=\frac{a_1}{a_2u^2}+\frac{b_1}{b_2uv}+\frac{c_1}{c_2v^2}. \end{equation*} Now we have \begin{equation*} f(ht,kt)_{-2} = \frac{a_1}{a_2 h^2t^2}+\frac{b_1}{b_2 hkt^2} + \frac{c_1}{c_2 k^2t^2} \end{equation*} \begin{equation*} = \frac{a_1b_2hkc_2k^2 + b_1a_2h^2c_2k^2 + c_1a_2h^2 b_2hk}{a_2b_2c_2h^3k^3t^2} \end{equation*} \begin{equation*} = \frac{a_1b_2c_2k^2 + b_1a_2c_2hk + c_1a_2b_2h^2}{a_2b_2c_2h^2k^2t^2},\end{equation*} and what we need is for $a_2b_2c_2|a_1b_2c_2k^2+b_1a_2c_2hk+c_1a_2b_2h^2$ (for all $h$ and $k$, of course). This is where I'm stuck. Reducing mod $a_2$ and recalling that $gcd(a_1,a_2)=1$ we should get that $a_2|b_2c_2k^2$, and similarly $b_2|a_2c_2hk$ and $c_2|a_2b_2h^2$. Putting $h=k=1$ this means that $a_2|b_2c_2$, $b_2|a_2c_2$, and $c_2|a_2b_2$. I tried breaking down these conditions into conditions on the prime factorizations but didn't get very far. And in any case, numbers that satisfy these need not satisfy the original relation. Help please????
Motivation: In case you're still reading and you're wondering why I care about this specific calculation, it's because it's closely related to the "$K$-theoretic homology of $K$-theory"; precisely, it's the image of $K_*K\rightarrow K_*K\otimes \mathbb{Q}$. This is important because for any (co)homology theory $E$, the $E$-cohomology of $E$ (written $E^*E$) is exactly the algebra of stable cohomology operations, and with some additional assumtions on $E$ (which are satisfied by $K$-theory), the $E$-homology of $E$ is the coalgebra of stable homology cooperations. Furthermore, the $e$-invariant is a homomorphism from the stable homotopy groups of spheres to a subquotient of $K_*K$!