I do not personally believe in literal omnipresence. I believe that God has a body of flesh and bones. I believe that the spirit and the body, inseparably connected, bring a fullness of joy. I believe that God is enthroned in yonder heavens. These things seem incompatible with literal omnipresence to me.
I am not opposed to God being functionally omnipresent. In a vague, general way I believe He is functionally omnipresent. But I have not given it much thought. How could an embodied God be functionally omnipresent?
That could be where an infinite number of intelligences/spirits come in. Joseph Smith taught that whatever it is that is ultimate to us - call it intelligence - was never created nor made, nor can be. This might suggest then that the number of intelligences is fixed (since God can't make more of them). But this number must be infinite, or there would come a time when God would run out of intelligences. But what do intelligences do with their eternal past?
Now, if we can imagine an infinite space, filled with infinite intelligences, we might conceive of a condition where every event in the universe could be directly observed by an intelligence. This observation could then be communicated to God. Thus an infinite number of intelligences could eternally be the 'eyes and ears' of God, and thus allow Him to be functionally omnipresent in spite of His flesh and bone embodiment.
Thus we would have a functionally omnipresent God, and a dynamic purpose and activity for infinite intelligences to engage in.