Before you go down the route of new DI boxes, mic's, stomp boxes, FX racks, re arranging the house and SSL desks,
try changing the Strings first, much cheaper.
The natural decay as you go up the neck on the D & g strings seems quite short and dead. [to me]
I had this problem a few years ago when for some reason i was using lighter gauge strings and a lower action.
Low action is not always desirable on Fretless, the higher notes won't sing out, and the low notes run into each other.
[ Different for each player, and Bass set up of course ]
I went back to cable gauge and all was fine again.
Garry