I question the assumption that any proposed god is implicitly necessary. You (and many others) are explicitly defining a god which is necessary. It is perfectly possible to conceive of an existence in which no god needs to exist for it to be internally consistent, therefore no god is automatically necessary.
I also see circular logic here. You are saying god is necessary if he exists but then god exists if he is necessary.
The rest is just unnecessary complexity. These two lines form the core of your claims. It doesn’t matter whether we (or anyone else) conceives of anything or not. The god you propose either exists or doesn’t.
As far as greatness goes, you can't be just compassionate but have to the best possible compassionate. Same with all his attributes like power, etc. So when it comes to existence, it has to the greatest type.
Of course, what you describe is a god, but not the God as in the one defined as not only the greatest but greatest possible.
If we can conceive of the greatest possible one, we can conceive the type of existence is not possible type, but necessary. And in fact, all arguments for oneness of God rely on this premise. That there can't be any god beside God relies that God is not just the actual greatest being, but greatest possible being. And further more, that his greatness encompasses existence to the extent nothing can exist with him. IF you can conceive of that, that means you have to conceive it's not the type that can exist or may not exist, it's the type no possible world can exist, but that he would be part of it, and it would not add to existence an iota of an amount of existence.
In fact another way to phrase the premise is to say, it's impossible if God exists, for anything to be independent of God. To prove this is simple. If any existence can exist without God, God would not be necessary. But if it's proven depends on God, not any possible existence in any possible world can exist without him, it's because he is necessary. And Necessary implies he is thus Samad and Al-Hayu and vice versa, which are two relate names in the Quran.
Oneness of God relies on the ontological argument.