Find the prime power factorizations of the two given numbers. We get
$3240=2^3\cdot 3^4\cdot 5^1$ and
$3600=2^4\cdot 3^2\cdot 5^2$.
Let our unknown number be $n$. Because the LCM of $3240$, $3600$, and $n$ only involves the primes $2$, $3$, $5$, and $7$, we know that the prime power factorization of $n$ can involve no primes other than these.
So the only question is: how many of each?
Since the HCF of our three numbers is $36=2^2\cdot 3^2$, the highest power of $2$ that divides $n$ must be $2^2$.
The LCM has a $3^5$. Since the highest power of $3$ needed by our first two numbers is $3^4$, the highest power of $3$ that divides $n$ must be $3^5$.
Note that $5$ cannot divide $n$ since $5$ divides $3240$ and $3600$ but does not divide $36$.
Note also that since $7$ does not divide the first two numbers, the $7^2$ in the LCM must come from $n$.
It follows that $n=2^2\cdot 3^5\cdot 7^2$.
Remark: Your intuition about "not enough information" is reasonable. For example, if the HCF of the three numbers was $72$ instead of $36$, then the highest power of $2$ that divides $n$ could be $2^3$ or $2^4$, so $n$ would not be completely determined.