Mixolydian = Major with b7
That scale is very common in rock. It is not as 'happy sounding' as pure major.
Tips to getting the scalar passage in "The Immigrant Song":
1. Practice it slow, with a metronome... oh, I see you are already doing that... hmmm...

2. Practice it slow, with a metronome... hmmm... I think I am getting a little redundant here...
Finally, you guessed it... practice it slow, with a metronome.
Really - there is no better way to get a fast part down. That part is really not that complicated - ascending mixolydian scales, over and over again - but it is physically challenging to play it fast, with a pick. Especially if you are not a 'native pick player'.
But if you really buckle down and work those scales with that metronome, you will get it!
It's a great song, and I have always loved that part in particular! Lots of fun to play! Keep at it, you'll get it!
Do you have a 'regular fingering pattern for the mixolydian scale? If not, try this (1 finger per fret)
Code:
#1 MAJOR SCALE: 4-string, version 1 (Starting on the E string).
G ----|-----|-----|-----|-----|-----|-----|----
D ----|----1|-----|----3|----4|-----|-----|----
A ----|----1|----2|-----|----4|-----|-----|----
E ----|-----|----2|-----|----4|-----|-----|----
#2 MIXOLYDIAN SCALE (IMMIGRANT SONG SCALE): 4-string, version 1 (Starting on the E string).
G ----|-----|-----|-----|-----|-----|-----|----
D ----|----1|----2|-----|----4|-----|-----|----
A ----|----1|----2|-----|----4|-----|-----|----
E ----|-----|----2|-----|----4|-----|-----|----
#3 MAJOR SCALE: 4-string, version 2 (Starting on the E string).
G ----|-----|-----|-----|-----|-----|-----|----
D ----|-----|-----|----1|----2|-----|-----|----
A ----|-----|----1|-----|----2|-----|----4|----
E ----|-----|----1|-----|----2|-----|----4|----
#4 MIXOLYDIAN SCALE (IMMIGRANT SONG SCALE): 4-string, version 2 (Starting on the E string).
G ----|-----|-----|-----|-----|-----|-----|----
D ----|-----|----1|-----|----2|-----|-----|----
A ----|-----|----1|-----|----2|-----|----4|----
E ----|-----|----1|-----|----2|-----|----4|----
#5 MAJOR SCALE: 5-string, version (Starting on the B string - 2 octaves).
G ----|-----|-----|----1|-----|----3|----4|----
D ----|-----|-----|----1|----2|-----|----4|----
A ----|-----|-----|----1|----2|-----|----4|----
E ----|-----|----1|-----|----2|-----|----4|----
B ----|-----|----1|-----|----2|-----|----4|----
#6 MIXOLYDIAN SCALE (IMMIGRANT SONG SCALE): 5-string, version (Starting on the B string - 2 octaves).
G ----|-----|-----|----1|----2|-----|----4|----
D ----|-----|-----|----1|----2|-----|----4|----
A ----|-----|----1|-----|----2|-----|----4|----
E ----|-----|----1|-----|----2|-----|----4|----
B ----|-----|----1|-----|----2|-----|----4|----
If I had to hazard a guess, I would say JPJ is using the 2, 4, 1, 2, 4, 1, 2, 4 fingering (#2 - MIXOLYDIAN SCALE: 4-string, version 1 above).