You should probably read what's on the Initiative
https://citizens-initiative.europa.eu/initiatives/details/2024/000007_en
You can find representatives and members there, nothing is an "invisible army" as you depict. There has historically not been a specific need for donations because the thing you donate is your time and your vote on the initiative (if you live in the EU), it doesn't require payment.
See the website (https://www.stopkillinggames.com/) or join the Discord for a "private forum or group" as you describe. To see the past actions, or as you say "who is doing what with whom and when", see the previous actions taken: https://www.stopkillinggames.com/pastactions
For information to "refute existing and possible claims made by lobbying groups and naysayers", you should re-read the FAQ, which does specifically and exactly that: https://www.stopkillinggames.com/faq Make sure to scroll the whole page and read every response.