If you don't plan on having it save passwords, you can add the following to the .desktop file your using for the Exec line:
--password-store=basic
The line for my chrome setup for some public computers at work is:
Exec=/usr/bin/google-chrome-stable --password-store=basic %U
So you'd just need to replace the file path to your chromium path and it should work.
I've used a fair number of them from Amazon, they all have worked in any browser. I think I might have had to manually specify http and not https for setup though.