I found a post on Stackexchange which might be of use for you

I am guessing that your ICUSB2324I handles break signals a bit differently than the WCH34x adapter. If the provided post isn't of use for you, I advise you to test around a bit more. You could try to use DTR or RST for the red light and BREAK for another light for example and see if that works. If you are able to, check the actual data frames transmitted and see if you can spot anything unusual there (or different to the working adapter). There are many layers involved in the actual communication between the devices, so it is very hard to deduct the exact error without further information about what exactly is the unusual behaviour.

To help you getting started on how and what to test for your configuration, check out these two resources:

FT2232H Datasheet (Which is the chip used on your adapter)

D2XX Programmers Guide (For the mentioned chip)

It also seems like the chip comes with an integrated method to set the BREAK Condition. FT_SetBreakOn() on page 38, maybe check that one out and test around.