You don't have to set the wheel up for the profiler, the profiler is just a tool that allows you to configure feedback effect strength and wheel rotation and make profiles for each game. Maybe there is a way to reconfigure all the binds but you shouldn't do that.
As for the game, I don't have it and I don't know anyone who does, but two things:
1. Not all games support wheels, in fact very few do. You can use a wheel as a controller in a lot of games, but if the game wasn't built with wheels in mind you can't expect a good experience; things like no feedback, no analogue control, limited wheel rotation, certain buttons or axes not bing supported, etc.
2. Some simulators have separate mapping menus specifically for axes, I think War Thunder is one. If you try to map an axis to a bind that expects a key, it works digitally (as you're describing). Since War Thunder is playable with a keyboard it's possible to accidentally set up a flight stick so that all the axes are digital, you have to go into the axes menu to be able to set each axis as an analogue input. Maybe have a search for that.
So keep looking but don't be surprised if the answer is simply that the wheel isn't supported. Also, try emailing the sim's developer for help.
As an aside, I've just googled a bit to see if the wheel is supported and I notice you've asked the Logitech forum about this, also implying that people can learn from the sim and train to use a real forklift with it and hence improve their employability. That's somewhat inaccurate because even if a simulator could teach you how to drive a forklift, no one is going to hire you to drive a forklift if your experience is solely simulator-based. Most people learn and subsequently get licensed to operate forklifts on the job, sponsored by their employer.
I've not seen anything about this sim but I'm assuming it's another one of those bizarro sim-games aimed mainly at the German market. In fact, even though I'm at work I'm going to look it up, I bet the developer is Eastern European or German...
Edit: Yep, German. Also I'm not sure a game that doesn't detect collisions with other people in the warehouse is going to teach you what you need to know to drive a forklift. I don't think your wheel will work with it, either.