The egg is held in a protective sack, clenched tight when not in freefall.
It's unclear how long it took you to figure out that this dragon needed to be in freefall for insemination to take hold, but I'm sure it wasn't an easy time for you.
Her egg(s) are fragile, so they're very securely held in an internal pouch while she's walking around, in flight, or being... experimented on. This prevents any sperm from getting to the egg. However, while in freefall, her natural instinct is to maximize her surface area and slow her descent. Part of the way this is done is by unclenching the egg sack and expanding its volume to provide more surface area outside her body. If you've ever unclenched your own egg sac you'd realize it's a very exciting feeling (this is why drops on rollercoasters are fun). This will put her in the mood, so watch out.
So, naturally, you can push her out of an airplane and skydive with her. You don't even need special equipment (for the dive), 4000 meters is at the upper limit of elevation before a skydiver needs to wear supplemental oxygen. You'll also get about 70 seconds of freefall (assuming a human's terminal velocity) to do the deed, as it were, before you'll need to open your chute. You can wear and artificial tail with the appropriate apparatus installed and just hang on to her as you fall together.
If she's not agreeable to being pushed out of a plane or you don't trust that she'll come back home after what you did to her in freefall, you can just ride the vomit comet and keep her caged. The freefall without the wind should make your work a bit easier and will likely still trigger the (in)appropriate response in the dragon.