Can you not lie the new pieces over the old and glue them down using PU glue and screws, then reinforcing the joint with plywood, glued and screwed, over the old and new?
PU glue like Gorrilla glue that’s under £20 a litre from toolstation
I don't trust that Gorilla PU glue, it's just not as strong as they say. A decent d4 pva based glue is much better on joinery.