The global binds are removed at the end of the setup if they are not present on your character set. It is normal.
Someone who was having a similar issue (with the WoW PTR though) was resolved when setting WoW to load the 64bit client. It was found that when InnerSpace was loading the 32bit client, and then the client would load the 64 bit client (because they were on the 64bit OS), that the sessions would get all weird.
You have a couple of options as far as I can tell. Make sure that your game launcher is shutting down correctly before the next game is starting up (you can extend the time between character slot launches on your Character Set).
Also check to see that you are not getting a second game process start up. This should actually be quite visible in the Inner Space Console (right click the icon in the system tray, select Console), in the section where is says something like
- Code:
..<snip>...
| 17:45:00 ISBoxer: Launching Slot 1 Character 'dxNada'
Successfully executed '"C:\Program Files (x86)\InnerSpace\dxNothing.exe" {CHARNAME}'.
Session name 'dxNothing2164' accepted for process C:Program Files (x86)InnerSpacedxNothing.exe
Session name 'is1' accepted for process C:Program Files (x86)InnerSpacedxNothing.exe
Session launched (is1): "C:/Program Files (x86)/InnerSpace/dxNothing.exe"
| 17:45:05 ISBoxer: Launching Slot 2 Character 'dxRight'
..<snip>...
Grab your console log from a character set launch and post if here if you can see anything (strat from the top left of where you want to copy, left click the mouse, hold it down, then drag to highlight, when you let go it will copy the text to the clipboard).
Finally, you could try setting InnerSpace to 32bit, see if that makes a difference. You may need the Dev version. Right click Inner Space in the system tray (notification area) and select Patcher, select (red dot) "Download development (test) patches" and then close and restart Inner Space to let it patch up.
Once that is done, right click Inner Space in the system tray (notification area) and select Configuration, select (red dot) the "Use 32-bit Inner Space Uplink" option on the General tab.