It is an open conjecture from a paper of Konyagin and myself that every $n$-element set of integers is Freiman-isomorphic to a subset of $[0,2^{n-2}]$. There are some counterexamples for small values of $n$, but it is believed that the conjecture is "essentially true". If so, your $4^n$ actually can be improved to $2^n$ (and in fact, to $2^{n-2}+1$). The set $\{0,1,2,4,\dotsc,2^{n-2}\}$ shows that $2^{n-2}+1$ is a sharp bound.

The bound $4^n$ is easy, but the proof I can think of is too long to present here. You can find better estimates in Chapter 20 of the monograph Structural Additive Theory by David Grynkiewicz.

It is an open conjecture from a paper of Konyagin and myself that every $n$-element set of integers is Freiman-isomorphic to a subset of $[0,2^{n-2}]$. There are some counterexamples for small values of $n$, but it is believed that the conjecture is "essentially true"; if so, your $4^n$ actually can be improved to $2^n$ (and in fact, to $2^{n-2}+1$). No further improvements are possible: the set $\{0,1,2,4,\dotsc,2^{n-2}\}$ is not isomorphic to any shorter integer set. (If it is isomorphic to a set $\{a_0,a_1,\dotsc, a_{n-1}\}$, then $a_0+a_2=2a_1$, $a_0+a_3=2a_2$, ... , $a_0+a_{n-1}=2a_{n-2}$; assuming without loss of generality $a_0=0$, this results in $a_i=2^{i-1}a_1$, so that the diameter of $\{a_0,a_1,\dotsc, a_{n-1}\}$ is $|a_{n-1}-a_0|\ge 2^{n-2}$.)

The bound $8^n$ (somewhat weaker than you indicated) can be obtained as follows.

It suffices to show that if $A\subseteq[0,l]$ is an $n$-element integer set not isomorphic to a set of diameter smaller than $l$, then $l<8^n$. Write $A=\{a_1,\dotsc,a_n\}$ and fix a prime $2l<p\le 4l$, so that $A$ is isomorphic to its image $A_p$ under the canonical homomorphism $\Bbb Z\to\Bbb Z/p\Bbb Z$.

Consider the $n$-dimensional integer lattice $\Lambda:=(a_1,\dotsc, a_n)\Bbb Z+p\Bbb Z^n$. It is easily seen that the determinant of this lattice is $p^{n-1}$; hence, by Minkowski's First Theorem, $\Lambda$ has a non-zero point in the $n$-dimensional cube $[-p^{1-1/n},p^{1-1/n}]^n$. Consequently, there exist integers $y_1,\dotsc,y_n\in[-p^{1-1/n},p^{1-1/n}]$, not all equal to $0$, such that for yet another integer $z$ we have $y_i\equiv za_i\ (i\in[1,n])$. It is immediately seen that $z\not\equiv 0\pmod p$; hence, the set $\{y_1,\dotsc,y_n\}$ is isomorphic to $A_p$, and therefore to $A$. The assumption that $A$ is not isomorphic to a set of diameter smaller than $l$ implies now $2p^{1-1/n}\ge l$. Combining this with $p\le 4l$ yields the desired bound $l<8^n$.

For $l$ large, the prime $p$ can be chosen to satisfy $p=(2+o(1))l$, leading to $l<4^{(1+o(1))n}$. Much better estimates (something like $2^{(1+o(1))n}$, I believe) can be obtained using the method of the aforementioned paper; see also Chapter 20 of the monograph Structural Additive Theory by David Grynkiewicz.