Q: Is it possible to use a psalm as the 1st reading (like Psalm 23) followed by a sung psalm at the responsorial psalm time in a Catholic Funeral liturgy?
A: This is no provision for it. The psalm integrates into the liturgy in other ways, not as a replacement for one of the readings.
I see the dilemma, though. Psalm 23 has its own resonances among the faithful. It stands as an exemplar of the Word of God.
I guess another solution would be – after the first reading – to have everyone sing the psalm refrain and then have the cantor read the entire text of Psalm 23. That is not too different from the way it looks in the Simple Gradual.