How can I force the mobile app to upload photos?

The lack of actual support on this issue is ridiculous.
This isn't an uncommon or difficult to understand issue.
The phone app seems to only sync photos randomly even when camera uploads is on, the battery is full, wifi or even wired connection is present, there are no potentially conflicting apps, etc.

This does not require screenshots or random questions to comprehend or even reproduce.
Manual uploads or repeatedly turning camera uploads off & on again are not acceptable solutions.

This is an issue most immediately solved by simply adding a manual/force sync or refresh button on the app.

The larger issue almost certainly has to do with how Dropbox prevents itself from being a drain on battery & processing power of the phone. If it were to continually monitor for changes to the synced files & directories, it would necessarily impact data usage and potentially impact performance of the device and other apps. Thus it's set to only do so at some random interval or based on an assortment of other factors.

However, the objective of automatic backup is to prevent loss or avoid having to manually transfer items between devices. As such, adequately serving those functions will frequently require more rapid sync or, at minimum, the ability to turn on continual sync for a predetermined period of time. Similar functionality has long existed in the desktop version of the app.

As such, and like many other reported problems, actually solving the issue, avoiding further user frustration and complaints should be relatively easy. But that requires the issue getting past "Support" treating it like it's an unknown issue and otherwise mishandling support requests en masse.