The new $\| \Delta \|$, defined as $\mathop{\rm Im}(\tau)^6$ times the absolute value of the usual modular form $\Delta$, is invariant under the full modular group $\Gamma = {\rm PSL}_2({\bf Z})$ acting on the upper half-plane $H$. This $\| \Delta \|$ is nonzero and continuous on the quotient $H / \Gamma$, and approaches zero exponentially as $\tau$ approaches the one cusp of $H / \Gamma$. Hence $\|\Delta\|$ is uniformly bounded above, without any hypothesis on $j$; and $\|\Delta\|$ is bounded below if we have an upper bound on $|j|$. The latter bound is completely effective, namely $$ \| \Delta \| \gg (\log|j|)^6 / |j| {\rm\ \ \ \ as\ \ \ \ } |j| \rightarrow \infty, $$ and indeed $\| \Delta \| \sim C (\log|j|)^6 / |j|$ for some universal constant $C$, which is $(2\pi)^{-6}$ if I did this right. Now if you bound the height of $j$ from above then you impose an upper bound on the absolute value of any conjugate of $j$, and thus on $\| \Delta \|$.
Whether and how this lower bound depends on the height of $j$ then hinges on which flavor of height you're using, i.e. whether you normalize according to the degree $[{\bf Q}(j) : {\bf Q}]$, and whether you take logarithms. There is no such bound in the other direction: large height of $j$ does not force small $\| \Delta \|$ because it does not force $j$ to have a large conjugate (e.g. $j$ could be $1 / 10^{100}$).