I believe they have one program for all their keyboards. Tipro are the same: one program for all their keyboards.
Yes … and no! From a closer reading of some of their datasheets, what you cannot change is the hardware keycode generated by each key. Hardware uses these raw keycodes to decide which physical key has been pressed; as you correctly state, the keycode produced by pressing each key is fixed on the TA61, whereas more expensive models in the range do allow you to do this. This is equivalent to physically moving (for example) the physical E key to a different place on the keyboard, and is a very low-level way of altering what the keyboard does. Unless you are writing device drivers, this isn’t what you want to do anyway.
BUT (and it is a very important BUT!) what you can program is the keystroke or keystrokes which each function key on your TA61 (‘46 freely programmable function keys’) sends to the Windows keyboard buffer when that key is pressed. For example, the function key which normally sends E can be programmed to send Ctrl+Shift+F5 instead. Which is exactly what you want to do!
My understanding is that this is what the Keyboard Programming Utility does, and I can’t think what else the phrase ‘46 freely programmable function keys’ (from the Nixdorf site) could mean? You can’t change what the 14 numeric keys do, but you can change what the 46 ‘function keys’ do.
If I were you, I would download the utility and try it anyway. At worst, it will tell you the keyboard cannot be programmed (in which case, I’m wrong); at best, you can set up your TA61 to work with mAirList. You could also e-mail or call Nixdorf and ask them: I find these big companies are usually happy to help.