Has Moses entered the kingdom of God, or is he at the gate looking in?

Moses has not entered the kingdom of God. He died and will remain dead until the resurrection. Moses is mentioned in Hebrews 11:23-24 in the list of faithful people. At the end of that chapter, we are told

And all these, though commended through their faith, did not receive what was promised, since God had provided something better for us, that apart from us they should not be made perfect. (Hebrews 11:39-40)

So Moses has not yet received the promised kingdom of God, but will do so at the same time all faithful people enter the kingdom.

Jesus said

Truly, truly, I say to you, an hour is coming, and is now here, when the dead will hear the voice of the Son of God, and those who hear will live. … Do not marvel at this, for an hour is coming when all who are in the tombs will hear his voice and come out, those who have done good to the resurrection of life, and those who have done evil to the resurrection of judgment. (John 5:25-29)

Moses will be one of those who will hear the voice of Jesus and be raised to life. Then he will enter God’s kingdom.