Whistling can be one of those tricky things to record.
Get right up on the mic, whistle "out," and you may get massive "breath 'plosives" (not technically "P"'plosives, but still air rushing directly into the capsule.
Move farther back (remember that most whistling is pretty quiet relative to most singing), and induce your room more into the picture, with a corresponding lessening of the whistle fundamental, plus potential for increased noise from your preamp chain.
If The Whistler can do so, a great remedy is to whistle "in" rather than out...thusly the breath is directed into the mouth, rather than into the capsule.
Your solution was a pretty good one, but your concerns are valid.
Pop screens, especially the nylon or metal perf ones, may cause some unnatural high frequency excursions, as comb filtering comes into play.
Foam screens will sound more natural, but have less 'plosive-reduction effect.
Listen to some old Gene Pitney records, such as "Only Love Can reak A Heart" to see how whistling is done right. Gene was the NYC "session whistler" for years.
Worst case, a stage microphone might have better inbred wind protection.
Interestingly, just this week I was remembering and discussing the recording of the whistle on "Dock Of The Bay" with Steve Cropper.
Good luck.