Create "Fake keys" that the computer interprets as real?

When you press a key on your keyboard, the hardware generates a 'scan code', which software then interprets as a certain key press.

Key reassignment

KeyTweak allows you to reassign the key press that Windows interprets from each scan code by adding appropriate Registry entries.

Unfortunately, it doesn't allow you to add additional fake keys. The closest it gets is allowing reassignment of the media/web control keys that some keyboards have. It also doesn't let you assign key codes that your keyboard doesn't normally use. So if you have a US keyboard, it won't let you assign Ó or £. So you would still be limited to using the normal key letters, which you say you've exhausted.

Faking it

AutoHotkey allows scan codes and unusual symbols to be generated by scripts.

This script generates £ whenever Win+p is pressed (I got the symbol off the internet, and pasted it into the text editor I was using to create the script):

#p::
send, £

Depending on your game, you might be able to use a script like this to get £ as a key in your game. But, if the game is waiting on actual keyboard scan codes to be generated, this won't work. It depends on how the game interacts with hardware.

Luckily, AutoHotkey can also generate keyboard scan codes. This script will generate a Space when Win+p is pressed, because scan code 39 (hex) is space (see Scan Code List).

#p::
send, {sc39}

AutoHotkey also has another method for generating Virtual Keys. I haven't researched the difference, but this script will generate space using a virtual key when pressing win+p:

#p::
send, {vk20}

This reference lists all the Virtual Key codes, including many non-standard keys that you won't have used yet: https://autohotkey.com/board/topic/98757-how-to-create-a-new-virtual-keys/

You could try to create a script that sends a non-standard Virtual Key to your game. If the game registers it and lets you save it in its keyboard set up page, the you are probably going to get this to work.

The last piece of the puzzle is getting the Logitech macros to call the AutoHotkey script that generates the Virtual Key. You'd create an AutoHotkey script without a hotkey, so it fires straightaway when run - such as

send, {vk7B}

or

send, £

Associate *.ahk files with the Autohotkey program, and then get the Logitech macro to open the saved script.

Note: I haven't tested this in any games. It will likely have different results in different games - depending on what they accept as valid entries. In the Virtual Keys link above, people are discussing doing almost exactly what you want to do, and seem to have had some success with using F13-F24 keys, so maybe try those first.

vk7B    sc58    F12
vk7C    sc64    F13
vk7D    sc65    F14
vk7E    sc66    F15
vk7F    sc67    F16
vk80    sc68    F17
vk81    sc69    F18
vk82    sc6A    F19
vk83    sc6B    F20
vk84    sc6C    F21
vk85    sc6D    F22
vk86    sc6E    F23
vk87    sc76    F24

EDIT: As requested in comments; a simple script to create F13 could look like:

b::
send, {vk7C}

Every time you hit 'b' while this script is active (in your tray near the clock), F13 will be sent. This script will stay active and run every time you hit 'b' until you kill it. Once the game registers F13 in key set up, kill this script, and make one like this for Logitech to call:

send, {vk7C}

This one won't stay active, it will run once and then end, each time it is called.

Apologies for the late response. I had a need to access the function keys past F12, something current keyboard manufacturers don't seem to account for, but easy enough with AutoHotkey. I've used this to add these keystrokes as macros to my mouse, which doesn't support adding arbitrary keys otherwise.

z::
send, {F13}
return

x::
send, {F14}
return

c::
send, {F15}
return

v::
send, {F16}
return

b::
send, {F17}
return

n::
send, {F18}
return

m::
send, {F19}
return

l::
send, {F20}
return

k::
send, {F21}
return

j::
send, {F22}
return

h::
send, {F23}
return

g::
send, {F24}
return