Ok, the port settings in the Windows device manager are of no consequence. Don't waste time there.
The baud rate etc. is controlled by the LOGic setup. Are you sure that the radio is set to 38400 baud? Don't trust the manual. Most rigs allow you to change the baud rate, so go into the rig's menus and check it. (you said 384,000 in your post -- I trust that is a typo hihi). Try 1 & 2 stop bits.
If you're sure the COM port and baud rate are OK, the next thing I would do is try it with OmniRig.
http://www.dxatlas.com/OmniRig/
If it works with OmniRig, then we need to take a look at LOGic. I'm pretty sure we have people using the FT1200, but am not positive.
LOGic interfaces to OmniRig, so if you get it working with that, it will work with LOGic also. Ideally it would connect directly to LOGic, but in the meantime....
Let me know what you find!
73,
DH