I spent all day looking for this, most google results give me the N64 ROM injection articles.
Here's a list of all IOS that Goldeneye Wii updates with the disc update.
ios10 v768
ios11 v256
ios12 v269
ios13 v273
ios14 v520
ios15 v523
ios16 v512
ios17 v775
ios20 v256
ios21 v782
ios22 v1037
ios222 v65280
ios223 v65280
ios249 v65280
ios250 v65280
ios254 v260
ios28 v1550
ios30 v2816
ios31 v3349
ios33 v3091
ios34 v3348
ios35 v3349
ios36 v3351
ios37 v3869
ios38 v3867
ios4 v65280
ios50 v5120
ios51 v4864
ios53 v5406
ios55 v5406
ios56 v5405
ios57 v5661
ios60 v6400
ios61 v5405
ios70 v6687
ios9 v778
EDIT: Reason I did this is because I have a retail copy of the game, not a "backup" copy
EDIT2: With physical copies, the IOS files need hand-updating to preserve homebrew
Here's a list of all IOS that Goldeneye Wii updates with the disc update.
ios10 v768
ios11 v256
ios12 v269
ios13 v273
ios14 v520
ios15 v523
ios16 v512
ios17 v775
ios20 v256
ios21 v782
ios22 v1037
ios222 v65280
ios223 v65280
ios249 v65280
ios250 v65280
ios254 v260
ios28 v1550
ios30 v2816
ios31 v3349
ios33 v3091
ios34 v3348
ios35 v3349
ios36 v3351
ios37 v3869
ios38 v3867
ios4 v65280
ios50 v5120
ios51 v4864
ios53 v5406
ios55 v5406
ios56 v5405
ios57 v5661
ios60 v6400
ios61 v5405
ios70 v6687
ios9 v778
EDIT: Reason I did this is because I have a retail copy of the game, not a "backup" copy
EDIT2: With physical copies, the IOS files need hand-updating to preserve homebrew