GSD choice is right, encourage your girls to keep calm and collected as they go upstairs, so it doesn't become a game of tag!
Since you've definitely ID'd the girls as being more of a trigger, and that he definitely understands that the upstairs is off limits, that makes your job easier.
Though he is obviously the most "up/amped" when your girls do it, that's a higher stimulation/reaction. Does he show any change of behavior with you or your son? If he does, correcting the behavior with a "trigger" that is lower stimulation will be an easier first step. If it really is just the girls, I'd suggest to keep doing what you're doing, with a few tweaks.
Set up the stair passes as you've been, but be sure you start with Klaus in an actually relaxed/calm state. (which means no treats right away - if he's at all food motivated, that will up his level of brain engagement) Once you know he's calm, do some low level pass-bys as well as the other stuff you've been doing. As long as he stays calm/relaxed/in his down (or whatever behavior you're setting up), he can have some treats (but make the treat delivery low key, it's all about maintaining the calm mindset). As soon as he "passes" that level of challenge, up the ante and make the activity more intense, still only rewarding as he stays calm. If he starts to get amped up, try a firm "no," and continue as long as he comes back down to a low level of energy. If he continues to amp up, regress your level of stimulating behavior, and keep building on it as he adapts.
Doing this, you'll hopefully be raising his threshold for stair interactions, making him less likely to react. As you're watching for any "upping" of response level, give him a verbal "no" or "hey" with any change you see. If he starts in a calm down, but then gets alert (even holding the down) that's going from level 1 to level 3. Don't let him amp up to level 4 or 5 without a correction/redirection. If after the "hey" he goes back down to a 1 (or 2), THAT'S when you give the treat. (again, in a nice, calm manner)
Hopefully that makes sense? I just speed-typed this all and did zero proof reading. I used this method for desensitizing my keeshond to doors being opened (by anybody), and it made a huge difference.
It may also help to train a "chill out/cool it" behavior away from the stairs, and then use it i that context after Klaus learns to generalize it to many situations. (this type of behavior is AWESOME and useful in so many situations.)