You do not have to shut down Robot in order to use Functional Tester and vice versa, which includes recording sequentially.
TestManager enables you to view logs generated by different test case implementations on the suite level. Therefore, Robot and Functional Tester scripts will generate separate logs but these can be viewed together on the suite level through TestManager.
If you mix Robot and Functional Tester scripts in a suite and run the suite on a single computer under TestManager, a single log is created. If the Robot and Functional Tester scripts run on different computers under TestManager, each will have its own separate "computer" log, but both of these are accessible from the single log for the suite.
Note About TestManager Integration: Functional Tester is integrated with Rational TestManager version 7.0.1.2. If you have the 7.0.1.2 version of TestManager, you will be able to use the integrated features of Functional Tester and TestManager. See Understanding Functional Tester integrations for important information on using the products together.