Yes, bleach goes to completion, but it forms Iron II salts. The pH and the amount of excess EDTA or NTA is critical to prevent dye hue problems and to prevent Iron II stain from forming in the coating. So, the E6 bleach is designed to follow (more or less) a pH 11 developer, while the C41 bleach is designed to follow a pH 10 developer. After these, a pretty routine near neutral fix is used, adjusted for the pH differences of the two processes.
The E6 bleach is also designed to work with a pre-bleach.
Along with this is the need to bleach high iodide emulsions with total silver developed in E6 and only about 50% of the silver developed in C41 but with DIR couplers creating interfering chemistry.
It can be done, but in the case of E6 films, you may see a higher dmin, generally tending to be yellowish, and in C41 films and E6 films you will tend to see desaturated colors, higher grain and lower sharpness. That is, if you don't take the bleach to completion. So, you have to test to find out what is completion, and when you reach it, you have to determine if you have harmed the final negative or slide. Once you find the correct conditions, you can probably then process safely. That is, aside from any dye stability issues that might arise.
Since the change to Bleach III, IDK what that would mean for E6 films, nor do I know what to recommend for C41 films in E6 bleach. I suspect there will be few problems, but I don't just want to say OK go ahead! That would be irresponsible of me. I have to show you both sides of the issue.
PE