The inversions are fair play. Learn them thoroughly. The quicker you get at identifying them the better.
See the Maj7 Lydian Chart below!
Cmaj7 - C E G B
C7 - C E G Bb
Cmin7 - C Eb G Bb
See if you can apply these intervallic structures to the chart below. Iv'e already mapped out CMaj7/Lydian with scales for you :) Good luck!