Before any attempt at answering this question, one should address the unfortunately too common misconceptions that comes from those neologisms that are in appearance easy to understand. Namely, Being-with-others refers to an existential characteristic of the Dasein and not to an existentiell description (i.e. a description of how the Dasein lives).
Here is the relevant excerpt (SuZ, §26, p.164) followed by the translation of Stambaugh (correcting usages of "Da-sein" that should read "Dasein", a strange choice from Stambaugh):
Nach der jetzt durchgeführten Analyse gehört aber zum Sein des Daseins, um das es ihm in seinem Sein selbst geht, das Mitsein mit Anderen. Als Mitsein »ist« daher das Dasein wesenhaft umwillen Anderer. Das muß als existenziale Wesensaussage verstanden werden.
But, according to the analysis which we have now completed, being-with-others belongs to the being of Dasein, with which it is concerned in its very being. As being-with, Dasein "is" essentially for the sake of others. This must be understood as an existential statement as to its essence.
Here we see how speaking of being-with-others, and how it relates to the being that the Dasein is, immediately sets the discussion in the domain of existantiality. Whence the next question: what is meant by "existential" and "existentiell"? Heidegger writes (SuZ, §9, p.59):
Alle Explikate, die der Analytik des Daseins entspringen, sind gewonnen im Hinblick auf seine Existenzstruktur. Weil sie sich aus der Existenzialität bestimmen, nennen wir die Seinscharaktere des Daseins Existenzialien.
All explications arising from an analytic of Dasein are gained with a view toward its structure of existence. Because these explications are defined in terms of existentiality, we shall call the characteristics of being of Dasein existentials.
The capital term here is "structure": existential refers to the structures of the Being (Sein) of the Dasein (hence on the ontological side) that express themselves through the existentiel constitution of Dasein as being (Seiende), hence on the ontic side. This distinction should not be conflated with the essence and existence. Indeed, observe that most of the essential feature of a Dasein (having blue eyes, being friendly as a general trait) are existentiell as they can only be ontically meaningful. However, being friendly (or being a recluse) is only possible as an expression of the existential (hence structural) trait of being-with-others, which could also be seen as an essential (ontological) feature of the Dasein. As an illustration of this structural considerations, we could consider what we commonly call "objects". Objects all have some kind of materiality and this is actually a structural and ontological property of objects, as this materiality could be part of the definition of what an object is. Still, this property expresses itself ontically in many different way, taking for example a chair and a sheet of paper.
Further with this example, we could ask ourselves as you did, whether there is an antonym for the structural property of objects that we call "materiality". Obviously, we are tempted to say "immateriality", but there is something very unsatisfying with this answer as "immateriality" is not a structural property of objects and, as such, in the ontological realms of objects, does not mean anything. On the contrary, if we were to look for the antonym of an ontic notion for object like "heavy", we would be satisfied with "light", because it is also an ontic notion that applies to object (in virtue of the structural property of having a weight).
And same goes for the question, which now more precisely reads: "Does the existential property "Being-with-others" have an antonym?" In the domain of the existential structure of the Dasein, there can't be an antonym for it, as this very property determines the existential domain in which it is meaningful and so, where it couldn't coexists with its antonym (which would be a contradictory existential property). In a way, this is simply a more elaborate way to say, as you noted, that "[Dasein] has Being-with-one-another as its kind of Being". And conversely, "Being-with-others" is only meaningful in the context of the existential structure of the Dasein, so we could argue that, strictly sticking with the notion as developed in SuZ, there altogether can't be an antonym for it.