The inequality you want to prove

|K|^{\frac{n-1}{n}}\le \frac{1}{n,\omega _n^{1/n}};H^{,n-1}(\partial K) \qquad (1)

for every compact set K ⊂ ℝⁿ is nothing other than the usual isoperimetric inequality written with the geometric – i.e. Hausdorff – surface measure instead of the Minkowski (outer) perimeter μ₊(K).
At first sight it therefore looks as if one should only have to show

μ₊(K)\le H^{,n-1}(\partial K) (2)

and then insert (2) in the Brunn–Minkowski version

|K|^{(n-1)/n}\le\frac{1}{n,\omega _n^{1/n}};\mu _+(K). (3)

Unfortunately, inequality (2) is already a highly non–trivial fact; in fact it is equivalent to one of the deepest basic theorems of Geometric Measure Theory, namely

(*) If H^{,n-1}(\partial K)<\infty then K has finite De Giorgi perimeter (P(K)).

Recall that De Giorgi’s (or BV-) perimeter is defined by

P(K)=\sup\Bigl{\int_K\operatorname{div}\varphi,dx:\varphi\in C_c^1(\mathbb R^n,\mathbb R^n),;|\varphi|\le1\Bigr};

for smooth sets it coincides with the (n−1)-dimensional Hausdorff measure of the boundary. Federer proved in 1959 that

H^{,n-1}(\partial K)<\infty \Longrightarrow P(K)\le H^{,n-1}(\partial K);

this is exactly (2) (the outer Minkowski content is another equivalent definition of P(K)). Federer's proof (and all later proofs) uses

• Vitali–Besicovitch covering,
• differentiation of measures,
• the co-area formula,
• rectifiability theory,

i.e. the standard machinery of GMT.

Once one has (*), the desired inequality (1) follows in three lines:

finite H^{,n-1}(∂K) ⇒ finite perimeter P(K) (Federer) ⇒ |K|^{(n-1)/n} ≤ c_n P(K) (isoperimetric inequality for finite-perimeter sets) ⇒ |K|^{(n-1)/n} ≤ c_n H^{,n-1}(\partial^*K) ≤ c_n H^{,n-1}(\partial K).

Conversely, if one could prove (1) for all compact K in some elementary way, then for every K with H^{,n-1}(∂K)<∞ one would get a quantitative bound on |K| in terms of the outer surface increase of K; this can be turned into a bound on P(K) and hence would give a new elementary proof of (*). So an elementary proof of (1) would amount to an elementary proof of Federer’s theorem – something that has resisted all attempts so far.

In short:

• For smooth (C²) or, more generally, finite–perimeter sets inequality (1) is easy and classical.

• Extending it to all compact sets is equivalent to showing that “finite (n−1)-Hausdorff boundary measure ⇒ finite perimeter”, which is one of the corner–stone results of GMT.

• Every known proof of that cornerstone result uses the full GMT toolkit, and no substantially simpler argument is known or expected.

Therefore there is no genuinely elementary proof of inequality (1); to establish it one has to develop (at least a fair amount of) the theory of sets of finite perimeter – exactly the route you already mentioned in your question (Ambrosio–Fusco–Pallara, Ch. 3 and 4).