Unless you are racing, the 44 idfs will be smoother on the bottom to top transition and have better low end response. I have them on a 6.3 liter small block Chevrolet and they are just fine. I'm giving up 2-3% of the available horsepower on the top end by not switching to 48s, but the overall driveability is worth it.
Weber tuning is all about interrelated components:
Venturi sizing/jet sizing/emulsion tube sizing combined with fine adjustments of air screws. Unless you are prepared to study extensively and go through a protracted trial and error phase, using an experienced tuner is worth the expense.
44s are more than adequate for a 5 liter. Both setups will work. It all comes down to what you want and what you intend to do with the car.