I think I found a solution myself after thinking a bit more about my problem. For those interested:
assigning the current position of joint frames as parameters and using those in inequality constraints is obviously a bad idea, since the optimizer does not know about the dependency of system states (in this case configuration) of those parameters!
What should work would be constructing an inequality function for each collision pair (eg. link 1 and 4) which calculates Lumelskys Algorithm based on the system states. No need to adjust or extend the system state vector x, just using it as input to those inequality constraints.
Sorry for causing some confusion!
Best regards