Looks like a family heirloom in progress.
Have you considered a lock mortise chisel? (AKA a Swan Neck Chisel)
Swan Neck:Lock Mortise Chisels.jpg
These two, 1/8" & 1/4" are very helpful cleaning out the bottom of mortises.
USN 3:4%22 Chisel Mortising +1:2%22 swan neck lock mortise.jpg
The large lock mortise chisel on the bench is 1/2". My recollection is it cost me a little more than $50 with shipping.
For a 1/2" mortise a 3/8" lock mortise chisel might be preferred so as to not mar the edges of the mortise.
jtk