There's no Microsoft official way to do that.
But unofficially - Use Rufus to install Windows To Go on your USB drive with your Windows 11 ISO.
Microsoft discontinued Windows To Go in 2019 so in theory a future Windows update could break those type of installs but for now they still seem to work https://en.wikipedia.org/wiki/Windows_To_Go
~~PS - If Lockdown Browser works in a virtual machine environment maybe you also have the option of installing Windows 11 in QEMU + Virt-Manager and just running a Windows VM that way. Works great under Debian.~~
EDIT: Doing a quick search it looks like Lockdown Browser actually tries to check for virtual machine usage, ugh. Guess they really want you to run a physicial system for that :/ https://support.respondus.com/hc/en-us/articles/4409604116123-I-receive-a-warning-The-browser-cant-be-used-in-virtual-machine-software-such-as-Virtual-PC-VMWare-and-Parallels
Rufus still lets you create Windows To Go installations on a USB drive. (unsupported by Microsoft as you mentioned)
I am curious about your DISM approach, are you just applying the Windows install image onto the USB? Or are you doing it the long way by imaging a normally installed Windows system first? I am very familiar with DISM but never tried applying an entire usable Windows install onto a USB drive that way.