Engaging the P-51s early on is easily a loosing proposal for the LW in early 1944, since that would locate them within the scope of P-47s when airborne, and within scope of many tactical fighters bombers when at the airbase, taking off and landing.
It is and the Luftwaffe knew this. Most importantly any fighters attempting to engage the escorts early render themselves unavailable to attack the bombers, which is exactly what the 8th AF would have liked.
The Luftwaffe had limited resources and had to marshal them accordingly. In June 1944 Luftflotte Reich had 472 operationally ready single engine fighter (from a total of 788 ) but by the end of July this had fallen to only 273 (of 460).
Cheers
Steve