I don't follow you. To be omnipotent is to be all powerful. So you can't be omnipotent 'and' constrained. For those constraints operate as limits on one's power - they wouldn't be constraints otherwise. And as reasoned reflection reveals, a being who was not subject to those constraints would be more powerful than one who was not. Yet, again by reasoned reflection, no-one can be more powerful than an all powerful being.
So, an omnipotent being is not constrained by the laws of logic. And indeed, those laws would have to be in that being's gift, for how else could the being escape being bound by them?
The point can be made in numerous ways. I mean, imagine that I - uniquely among us - exist of necessity. Well, now I lack a power that you have - I can't take myself out of existence. That's a real constraint, yes? I really can't take myself out of existence, whereas you can. What difference does it make to point out that as it is a true by definition that a necessary existent cannot not exist, I cannot not exist. That's to miss the point spectacularly. It's just to make a point about the concept of necessity, but it does nothing whatever to imply that I do not, in fact, lack a power.
Now just apply this to God. If you say of God - as you are trying to do - that God exists of necessity, then God is constrained. And you can play about with definitions all you like, the fact is going to remain that God is constrained. If you redefine omnipotence in such a way that this God, this God who cannot take himself our of existence, turns out to satisfy your revised definition, then all you've done is show that your definition of omnipotence is worthless and no longer refers to a being who is all powerful.
Look, your case is faulty in two directions. First, you can't 'get to' a necessary being. You're starting with contingent things, yes? But then you - or you on Avicenna's behalf - are then assuming that if something exists contingently it must have a cause of its existence. That assumption is essential - you can't get to the conclusion that a necessity existent exists without it. Yet that assumption is demonstrably false.
So, the way is blocked - you/Avicenna cannot get to your desired conclusion. You have not provided us with any reason to think that there exists a necessary being. You've just made a leap - you've leapt from 'exists contingently' to 'needs a cause for its existence'. But that's clearly a mistaken leap - that is, a leap that reason itself - logic - tells you is a mistake if you listen carefully. For something can exist contingently yet be uncaused, and something can exist of necessity yet be caused. So you have leapt in defiance of logic, and it is only by defying logic that you have reached the conclusion that a necessary existent exists - a conclusion that logic will also tell you is flatly inconsistent with God existing.
That's the other direction - in addition to not having shown a necessary being to be needed to stop the regress of causes, no necessary existent can be God. So your foundation is faulty, but so too is the building you've constructed on top of it.
God is 'not' a necessary existent, for God is all powerful and so can take himself out of existence if he so wishes - thus he can not exist.
There are other problems too (though what I've said above is decisive, I think). I mean, if God is constrained by logic, then logic is a curious force in the universe that exists independently of God, yes? So now you're positing a Platonic universe in which there is some Form of Reason that determines what else can exist and what that which exists can do. That's not a universe created by God, that's a universe in which God finds himself. Which is, of course, inconsistent with God being God if, that is, we make - as most religions do, I think - 'being the creator of everything apart from himself' - a defining characteristic of God.
Nothing I've said above is illogical. Nothing I've said above implies that I reject the laws of logic. And nothing said above implies that God himself defies them. That God can do things that defy them does not mean he actually is or has. So nothing I have argued above involves a rejection of logic. All we are talking about is logic's power - and you think logic has power 'over' God, whereas I think that logic itself - that is, reasoned reflection - tells us if we engage in it carefully that God, being all powerful, has power over logic.