A burner phone I sort of understand, but why not take pictures?

Just get a dedicated camera if you are worried about using a smartphone camera.

It all seems too extreme, though I don't know what kind of activities you get up to while travelling.

As for how to transfer images to your own server, there are several ways.

If you are planning on using an RPi and a flash drive as the server then it sounds like you are quite out of your depth.

At a minimum, your server should be using a proper hard drive or SSD for storage, they are far more reliable than a flash drive.

Then you need to consider your connection to the internet from the server.

There are ways to setup a server that can be accessed from the internet without exposing it directly to the internet, though they will require some kind of external service provider.

You can also expose the server directly to the internet, but if you are inexperienced working with networks and computer security I would avoid that.

If you are absolutely convinced of the need for this paranoid level of security, I would suggest using VeraCrypt, you can create an encrypted file, where you put some fake or less important files, then you can create a hidden volume in the aame file for the actual stuff you want to hide. One password opens the fake volume, another opens the hidden volume, no one can prove the existance of the hidden volume without the password.

If you however are an average Joe, this level of paranoia is far too excessive, it will make you stand out far more from the crowd and generate more attention from the authorities.

I apart from using proper storage on a server, I won't tell you what to do, I will however suggest that your behavior makes you stand out from the crowd and that your attempts to become invisible makes you far more visible.

I wouldn't worry about showing pics of my butthole to a border agent. The bigger problem would be if they take it from you while it's unlocked, they can clone the whole thing, access your linked accounts, and implant invisible spyware.

If your server is running Linux you could run Termux on your phone and copy the files to the server via SCP. You could do it periodically with a script or you could run the command one time before you were going to travel across a suspect border.

Termux has the added advantage of being a generic app, without any specific purpose. There could be a situation where you don't want to have a 'Photo Downloader and Hider 9000' app on your phone.

If you just want something akin to iCloud/Google Photos, Immich is a good project to get started with.

Syncthing running on both setup to push files one way perhaps?

Don’t bother unless your burner phone OS writes random data over deleted files. Otherwise you’ll give a phone full of your csa images to border patrol with a file table that reveals they exist with the most trivial inspection.

I don’t want any of my files to be backed up on some government server

This part would be best solved by installing an alternate OS (e.g. LineageOS) on your Burner. Or using a device whose OS is not doing that at all.

The rest - Immich I guess.

All our phones use Syncthing to keep the entire phone synced to home.

Sync jobs for photos runs on any data connection, and on battery - other jobs run while charging, or only on wifi.

Be sure to use Syncthing-Fork, it moves these sync options into each folder pair/job.

For the home server, get a mini PC - they're tiny, sip power like Pi, and provide a lot more flexibility.