The short answer is that there's nothing you can do apart from keep checking. I am not aware that status will get you better availability. Maybe for DMs but no chance for SLs and GOs.

The reality is that miles are not easy to use and do not be fooled by influencers. I'm sure many people here have hundreds of thousands of miles and couldn't find the optimal way to spend it. Across all airlines, there are limited redemption opportunities for LH J flights because they are the most desired products. There isn't unlimited inventory for customers to burn their miles when they desire. The worst culprit is Qantas, where going to space is easier than finding international J availability.

Focusing on AMS, CX only operates a direct flight per day. It is significantly harder when compared to busier routes such as LHR. Even for LHR, last time I checked there was no inventory for July. Not to mention, the summer months for European destinations are particularly competitive.

Good luck