Hi,
I did not use Bamboo but in order to run TestComplete/TestExecute tests that interact with the UI of the tested application, the following must be done on the target test machine:
-- There *must* exist interactive user session for the user on behalf of whom tests are planned to be executed;
-- TestComplete/TestExecute (TC/TE) must be installed on the test machine;
-- TC/TE must have access to the test code (it's OK if the test code is on some shared network resource);
-- Tested application must be installed on the test machine.
So your task is to ensure that interactive user session exists on the remote machine; launch TC/TE inside this session and command it (TC/TE) to execute your tests.
The most reliable experience that personally I had for this type of task was to use the Remote Desktop Plus (http://www.donkz.nl) utility to open remote desktop session on the remote computer (utility supports encrypted credentials, something that is not easy to get with the regular mstsc one) and to use psexec to start TC/TE in the required session. As a possible workaround, you may consider to create a scheduled task on the remote test machine and set this task to execute when the user logs in and command the task to start TC/TE and execute your tests.