Maybe it wouldn't be limited by our universe, but surely it would be limited in its own domain. Maybe that intelligence is pretending to be an eternal being, because of not knowing how it came to be.
I am not saying the following is the case, but....
Perhaps that depends on the definition of "eternal".
I have always existed is not the same as saying
I have always known that I existed, and
I AM is not the same as saying
I have always known that I AM.
An eternal being could be composed of that which simply "always" existed -but developed complex self-awareness and intelligence one step at a time from simple self-awareness and intelligence (simple interaction is simple awareness, etc.)
Once complex intelligence and self-awareness develops, anything previously not considered can be considered and determined to have been a certain way based on complete available information -similar to how we can determine much of what has been in the past even though we have incomplete information. An overall self-developing intelligence essentially composed of everything and able to interface with everything could determine that it had considered everything -whereas we are composed of a small portion of everything which has already developed/been developed, and have access to only a portion of available information.
Though I disagree with the statement above... "God cannot exist without the Universe to create him - "He" is a construct - a product of our evolution and experience." ....it is logical that God must be composed of something -must exist as an arrangement of something -or, more correctly -everything.
The universe is not required for God to exist, but that of which the universe is composed is some portion of that which has always existed -which is also that of which God would be composed.
Everything would essentially be the "body" of God -and that part of which that is arranged to process, remember, etc. would be his mind -the mind being able to affect and arrange the body.
Though the elements once did not exist, that of which they are composed existed before them. Logic -or law -was applied to that which existed to determine a new arrangement -and those new laws (which are by new arrangement) must be based on the most basic absolute law/absolute logic.
God is quoted in the bible as saying he is accountable to himself. His own nature -the nature of that which exists -he cannot deny. The logic thereof is the basis of all truth.
Though you are technically correct in saying that he is "limited" (paraphrasing) to the most basic nature of all that exists, that nature is what allows for infinite creation, absolute truth, etc.
For example... not even God is able to say that 1+1 is not 2. Some have wondered if God is a mathematician -but he certainly must be. If "God is one" (the sum of all things which cannot be created or destroyed), and any activity occurs -math ensues. If there was only one or no thing and no activity, math would essentially not exist (except 0 or 1 and none to consider them). As things can become arranged and reconfigured, math continues to represent that arrangement and reconfiguration.
Also.... though God has the ultimate power of decision, those decisions must (if things are to go well, and as determined) be based on truth -and certain decisions are essentially no-brainers.
For example... if one overall intelligence develops, becomes self-aware (looks in a mirror, as it were), likes being himself, etc. -self-replication/reproduction is a logical step.
Though it may seem like limitation, it is actually a firm foundation upon which and from which anything imaginable may be built. Again, some things must be built due to the most basic nature of all things -and happiness can be derived from that which must be built -but much beauty and joy is in the details which need not be of any particular description.
Basic needs must be met, but meeting those needs makes many awesome things possible.