Maybe if you use the daylight spell and cast it on something outside of the globe of invulnerability and bring the item with daylight cast upon it within 75 feet of the item with darkness on it so that the spell emanations overlap.
The daylight spell states that:
If any of this spell's area overlaps with an area of darkness created by a spell of 3rd level or lower, the spell that created the darkness is dispelled.
The daylight spell emanates for 60 feet, the darkness spell emanates for 15 feet and the globe of invulnerability emanates for 10 feet and therefore the darkness spell has 5 feet of its emanation outside the globe of invulnerability where it will be affected by the daylight spell.
The rest of the answer comes down to semantics and how the rules are going to be interpreted by the DM:
Any spell of 5th level or lower cast from outside the barrier can't affect creatures or objects within it, even if the spell is cast using a higher level spell slot. Such a spell can target creatures and objects within the barrier, but the spell has no effect on them. Similarly, the area within the barrier is excluded from the areas affected by such spells.
The daylight spell has been cast outside the globe and is not targeting anything inside the globe and is not trying to affect a creature or an object within the globe; instead it is trying to affect a spell that is emanating from within the globe but also exists outside the globe and since the emanations overlap outside the globe of invulnerability then the daylight spell will dispel the darkness.
However, the DM might argue that:
- it is affecting the object upon which the darkness is cast and rule given the second sentence "Such a spell can target creatures and objects within the barrier, but the spell has no effect on them." that there would be no effect; or
- given the final sentence, "Similarly, the area within the barrier is excluded from the areas affected by such spells." and since the object at the centre of the emanation it would be unaffected by daylight.
I don't find either argument compelling as the daylight spell specifically states that it is sufficient for the area of light to overlap with any area of darkness for it to be dispelled and the daylight spell does not need to overlap with the globe of invulnerability to be able to achieve that and the object emanating the darkness will still be the same object whether or not it has the darkness spell active on it.
If the DM did argue that the darkness emanation inside the globe was unaffected by the dispelling effect from the daylight spell outside the globe then I would still want to argue that the daylight spell is a higher level light effect and would suppress the lower-level darkness effect outside of the globe and restrict its area of darkness effect to only the volume inside the globe and allow the players to see the globe of invulnerability (and darkness) as a 10' radius sphere which would fulfil your criteria of "allow enemies to see the globe of invulnerability for a brief moment" (where that brief moment is the 1 hour duration of the daylight spell).