Excellent Idea! (MAYBE there's a loophole)
First of all, I'm impressed by this strategy! It's a great idea, and turns the tables on the Beholder extremely well. You might have some trouble targeting the Beholder from within the cloud yourself, but characters normally know where other creatures are during combat (the fact that the Beholder hovers may complicate this, but probably not unless the Beholder takes the Hide action).
There is a small potential loophole I noticed, but it's not ironclad (more open to interpretation).
Legendary Action & Lair Actions
The Beholder can use its eye rays with its Action during its turn, but must choose targets it can see. However, it can also use its eye rays in two other ways: one is with a Legendary action at the end of another creature's turn. This legendary action is described as follows:
Eye Ray. The beholder uses one random eye ray.
It can also use lair actions, one of which is the following (MM, p. 27, bold added):
Walls within 120 feet of the beholder sprout grasping appendages until initiative count 20 on the round after next. Each creature of the beholder's choice that starts its turn within 10 feet of such a wall must succeed on a DC 15 Dexterity saving throw or be grappled.
Note that neither of these descriptions requires the Beholder to see its chosen targets.
It's likely that the definition of the eye rays given in the Actions section of its description is meant to apply to the other (Legendary) rays as well. But this is ambiguously worded, and open to interpretation. Clearly, some parts of the description of the eye rays must apply (or else there would be no limit to the range of these Eye Rays, for example): exactly how much applies, though, is ultimately up to the DM.
Even if its lair and Legendary actions do not require sight, note that your strategy would pretty much halve the amount of opportunities that the Beholder has to target creatures directly (as opposed to lifting a heavy object above a character while they are in the Antimagic Cone, and dropping it). This makes this strategy an excellent idea in any case.
Antimagic Field will prevail
We can look at this from two points of view, both from the rules and from a purely RP perspective. Let's check the rules first.
The description of Antimagic Field states that
Spells and other magical effects, except those created by an artifact or a deity, are suppressed in the sphere and can't protrude into it. A slot expended to cast a suppressed spell is consumed. While an effect is suppressed, it doesn't function, but the time it spends suppressed counts against its duration.
and that
Spells and magical effects such as dispel magic have no effect on the Sphere. Likewise, the spheres created by different antimagic field spells don't nullify each other
However, the Globe of Invulnerability can be dispelled or suppressed, as it specifies that only what is inside it is protected, and not the globe itself. Therefore it stands to reason that when the two spells meet, Antimagic Field will suppress the Globe of Invulnerability itself, which will allow the field to affect creatures or objects within the globe.
If we look at it from a RP point of view, we can once again turn to Antimagic Field's description, which says of the area of the field:
This area is divorced from the magical energy that suffuses the multiverse.
We could then say that this area lacks the required property to sustain a spell, which would suppress the globe. This isn't a case of a spell being stronger than another, such as light spells vs darkness spells. This is an instance of a spell removing the necessary energy for another spell to sustain itself, thereby making it fail.
Best Answer
Antimagic field only suppresses areas of effect while they overlap with the field; they function as normal otherwise
The antimagic field spell description answers your question:
As the phrasing of the sentence I've bolded suggests, the area of effect of other spells and magical effects is suppressed only while they overlap with the antimagic field.
Those parts of an AoE outside the area of antimagic field are not suppressed (though I'm not sure how it works if the point of origin of a spell like darkness is within the area of antimagic field). Once either the other AoE is moved or you move (since antimagic field is centered on you) such that there is no overlap, there is no AoE in the field to suppress.
This interpretation is further supported by the spell's description of its interaction with creatures/objects summoned or created by magic:
As you can see, antimagic field doesn't dispel/end the other spell; it simply "suppresses" its effects while the field overlaps the effect of the other spell (or magical effect).