There is no direct relationship between being inauthentic for Heidegger and being indifferent to the subjectivity of others for Sartre. There are several indirect relationships between these two ways of looking at the world.
First, for Heidegger, we need to get clear on what authenticity is. Authenticity (Eigentlichkeit) is when Da-Sein is being honest about what Da-Sein is. (Unfortunately, I will omit citations because the person I'm most familiar with who is published on this topic is me -- that's not to say others haven't done it better or are more famous in general). And for Heidegger what this Da-Sein (this human?) has as its deepest possibility is that: Da-sein is one day going to die.
Authenticity for Heidegger is then living in light of the fact you are going to die and not being distracted by the din of ontology / technology or anything else (here we could look at Being and Time or "The Question concerning Technology" any of several later texts).
Inauthenticity, then, is any way of living that is not living-towards-death. It's anything that distracts us from this possibility.
Turning to Sartre, my understanding is that other than in a footnote where Sartre explains that this account in Being and Nothingness is all about bad faith -- there's no moral claim that you must choose to acknowledge the subjectivity of others. I might be mixing my Sartre up a bit on this point, but I take it that even in Being and Nothingness the point to be highlighted is that I have a fundamental existential choice in whether or not to be in relation with the other and to what extent I let them define me. In other words, I don't see any moral probative force for choosing relationality for Sartre.
Now, I've hit upon a dicey point though if we go back to Heidegger, viz., is Heidegger making a moral or ethical claim? Is authenticity a moral category? Let's skip that debate, because that wasn't your question, and answer it:
Heidegger's authenticity is "relational" insofar as its about Da-sein's relation to Da-sein's certain future (death), but other people are distractions from this task.
Sartre's ideas on how we relate to others point to relating to them as objects, utility relations, or defining relations. But depending on how late in Sartre's career we're talking, there's some question as to how strongly he believes we need to relate.
Now, on to some indirect relations, there's some common background here in Kant, Hegel, and Kierkegaard.
Both inherit a way of reading Kant where Kant's philosophy of mind is seen as a meat-grinder with concepts and where Kant's ethics is seen as a free will that is non-relational and single-handedly rational in orientation. Both reject this -- but in a sense Sartre upgrades the freedom part while downgrading the rationality and Heidegger downgrades the rationality and makes the mind's utmost point its mortality.
Then, both inherit Hegel's master-slave dialectic of relationality. Contrary to popular caricature, the idea in Hegel is A and B fight and struggle until they realize they are the foundations of each others identity -- which then grows into a larger social whole and community. This isn't meant to be a moment in history. It's a post-Kant reason recognizing that the city precedes the individual (from Aristotle).
Both Sartre and Heidegger want off this train. For Heidegger, the social whole is another part of "ontology" which hides "metaphysics"/ "Being" and hides Da-sein's true nature from Da-sein. For Sartre, it's completely possible to stop the Master-slave dialectic at an earlier moment -- i.e. where the master kills the slave or where the master hates that the slave gives him identity (peeping tom section of Being and Nothingness). (when something is wrong in this dialectic Hegel calls it bad faith -- a term of great import in Sartre).
Final stop Kierkegaard. Kierkegaard too has a relational account of the self in Sickness unto Death and they both appear to have read it. Kierkegaard's Angst becomes Heidegger's Angst. Sartre's bad faith looks like defiance in Sickness -- it's precisely staking a claim against the larger whole.