Login without DB UI intervention?

Richard has this correct. This is possible, though it is not recommended. 

The iOS Core SDK does use OAuth 1, and the generated token on the App Console uses OAuth 2, as you noted, so you can't plug that in directly. You'd need to instead implement and run the OAuth 1 flow once for your account. There was another thread where I helped another developer through this in some detail here:

https://www.dropboxforum.com/hc/en-us/community/posts/204110983-Is-linking-necessary-to-download-files-

The StackOverflow posts I linked to there should be useful too:

https://stackoverflow.com/questions/15014001/allow-dropbox-api-to-access-my-account-on-users-device
https://stackoverflow.com/questions/27834922/auto-login-dropbox-account-on-core-api-without-login-prompt

Again though, I should emphasize that this is not recommended. A malicious user could extract the access token from your app and use it directly, bypassing any access controls you may have attempted to enforce.