Most other features are explicit that they are set at the time of summoning, but the attack modifier is not specified
The aberration has a number of features that are explicitly determined when the spell is cast
At Higher Levels. When you cast this spell using a spell slot of 5th level or higher, use the higher level wherever the spell's level appears on the stat block.
The AC, hp, number of attacks, and damage dealt by the aberration are all determined at the time the spell is cast, based on the level of the spell slot that was used.
When you cast the spell, choose Beholderkin, Slaad, or Star Spawn. The creature resembles an aberration of that kind, which determines certain traits in its stat block.
Even if you later choose another kind, that will not change its traits.
The aberration has just one aspect that is explicitly based on your current state
Summon Aberration is a concentration spell. If you lose your concentration the aberration will no longer be able to manifest at all.
The aberration has two features that do not explicitly say whether they are determined at the time it is summoned, or are instead based on your current state.
As you mention, the attack modifier of the aberration is determined as "your spell attack modifier". In addition, if the aberration is of the Star Spawn kind, its Whispering Aura has a spell save DC based on your spell save DC. Notably, neither the single qualifying word "current" nor a qualifying phrase such as "at the time you cast this spell" is present in the description. Thus, it is unclear which of the two possibilities determine these features. Your DM will need to decide how to implement these.
Other spells and abilities do implicitly assume 'as the spell is cast' in their descriptions.
This question is about storing spells in a Shield Guardian, which says the spell's parameters are "set by the original caster". This answer to the question explains that this means they are set at the time of casting and don't change as the caster changes.
Both the invisibility and greater invisibility spells include the phrase a "creature you touch becomes invisible until the spell ends. Anything the target is wearing or carrying is invisible as long as it is on the target's person." When I first read this, I was confused, because when I read "Anything the target is wearing or carrying is invisible as long as it is on the target's person." I could not understand what the target could be wearing or carrying that was not on their person. The spells only make sense when you understand the implicit 'at the time of casting' and parse them like so:
A creature you touch becomes invisible until the spell ends. Anything the target is wearing or carrying at the time of casting is also invisible as long as it is on the target's person; if it is later set down it will become visible, and objects picked up after the spell is cast do not become invisible.
What I would do
Narratively, you invest energy in the creation of the form of summoned creatures, or in bringing to your location the most powerful form you can, but once you have finished the summoning, they act as independent creatures under your direction. While you need to maintain your concentration on them to preserve their existence, their form itself has already been determined. It makes no narrative sense for the aberration's physical form to be continually 'checking in' with your state and then adjusting accordingly how well it can attack or how difficult its psychic energy is to resist.
Given the preponderance of features in the summon aberration spell that are explicitly determined at the time the spell is cast, the number of other spells and features that implicitly assume 'at the time of casting' in their descriptions, and in the absence of counter-examples of summoned creatures whose abilities are tied to the current state of their summoner, it seams more reasonable to assume that the two features not explicitly listed are also determined at the time of casting as well.