Weil's book L'intégration dans les groupes topologiques et ses applications

Leopoldo Nachbin's book "The Haar Integral" has Weil's proofs of existence and uniqueness of Haar measure, as well as Cartan's.

Weil establishes the Pontryagin duality theorem by an argument very similar to the original one by Pontryagin (in the compact/discrete case), which was extended to more general groups by van Kampen. Duality is established for $\mathbb{R}$, compact groups and discrete groups, and then generalized using structure theory of locally compact abelian groups. Hewitt & Ross has a proof along these lines, as does the second edition of Pontryagin's own Topological Groups (which was translated into English). Van Kampen's original paper is in English and available online, but it's not the easiest to follow.

It seems that Bourbaki has integrated at least some of Weil's ideas, and all of Bourbaki has been translated, as far as I know.

I really like Cartan and Godement's Théorie de la dualité et analyse harmonique dans les groupes abéliens localement compacts (1947). It isn't quite what you hope for, since it is again untranslated. But it is very much in Weil's spirit — improved to avoid the structure theory, but unlike Bourbaki's Théories spectrales, still eschewing Gelfand theory (replaced by use of the Kreĭn-Milman theorem).

The French is more accessible than Weil's, and it's much shorter: just 20 pages to progress through the theorems Riemann-Lebesgue, Bochner, Fourier inversion, Plancherel, and Pontryagin duality. Add two pages for Godement's generalized Stone's theorem (1944), and all the bases are covered.