@TerryTao In ['Addictive Number Theory', pp. 1–8 in: D. Chudnovsky and G. Chudnovsky (eds.), Additive Number Theory, Springer, 2010], Nathanson himself writes, "My Rochester articles were on a variety of topics, for example, [...] a result, sometimes called the “fundamental theorem of additive number theory,” about the structure of the iterated sumsets $hA$ of a finite set of integers [27]." Here, [27] is Nathanson's 1972 paper cited in the OP. Note the use of the definite article before "fundamental theorem".

@Keith In the proof of their 2nd claim, David Gao is using that $f(a \land b) \leq f(a) \land f(b)$, not that $f(a \land b) = f(a) \land f(b)$. This follows from $f$ being isotonic, plus the fact that $a \land b \leq a$ and $a \land b \leq b$.

Just noted a related thread on Math SE: math.stackexchange.com/questions/2897680

+1. I am accepting Emil Jeřábek's answer since it came first. However, I like this answer even more: it makes the proof neater and demonstrates that, after all, power set lattices are not that special with respect to the question raised in the OP.

Let me just add for my future self that $f(S) = T$: being isotone and surjective implies that f maps the max of the power set lattice of S (that is, S) to the max of the power set lattice of T (that is, T). So, $T = f(S')$ yields by injectivity that $S'=S$.

I guess you mean $S' \subseteq S$. Apart from that, awesome!