I believe you’re making this much more complicated than it needs to be. By its very nature, a mantray?na gives us a sacred object upon which to concentrate, namely (in our case) the Name of God.
But we don’t need to “narrow our consciousness by an intense concentration”, to use your formulation. For one thing, this sounds too much like thinking about what your fingers are doing, and not about the music, when you’re playing the piano, and it represents a significant departure from the tariki or “Other Power” sacramentally contained in the Name. One of the goals of this practice, after all, is to escape the illusion that everything somehow depends on us—on us who are not. So no: We should keep our attention on the mantram, not on our attention on the mantram, and we need not concern ourselves with the “degree of intensity”.
Perhaps the following from Schuon will be of some help:
“What matters a priori is not that we know how to concentrate; what matters is that we love to practice the Invocation…. It is better to invoke with joy while being a little distracted by harmless thoughts than to invoke without joy because the effort of concentration prevents one from being happy. It is necessary to guard against a perfectionism that is angry and ambitious, and basically individualistic; it is necessary to guard against all ‘zeal of bitterness’. It is better to invoke with carefreeness, like a bird which sings or like a child at play. Holy carefreeness readily combines with the sense of the sacred, thanks to confidence in God. Metaphysical knowledge and holy childlikeness must go hand in hand: ‘extremes meet’. A had?th says that there will be a people who will enter Paradise like a flock of birds” (“The Book of Keys”, No. 940, “Concentration”).
Unless I’m mistaken, what your Dzogchen teaching calls the “natural state” is what Schuon here means by “holy childlikeness”. But this is not opposed to concentration, nor a fortiori to the Invocation; on the contrary, it is this concentration in its purest form.