I think there's a clue in the detail of what Klopp says here:
"I am not careful, we get told what is possible and what isn’t. That’s it. In this range we try to work it."
That reads to me like he gets given a budget to work with (that is what he means by "range"). Within that budget FSG are probably happy for him to do whatever he wants, they'll trust his judgment on players and what he needs, but they don't want him to spend beyond the amount allowed. Whether you like it or not, this is not abnormal. The manager has discretion to shape his squad how he sees fit, but he doesn't have carte blanche to spend money the club doesn't have.
He also made a comment the other day about wanting to take more risks - I saw that as more likely to be a criticism of the transfer committee than the funds available to him. He maybe thinks the committee approach is too cautious. But with all of this, we need to remember that, when he signed, he said he had final say on transfers - I took that to mean that if the committee was recommending signing a player and he didn't want to do it, then we wouldn't sign that player. I'm not sure it necessarily means that if he wants to sign a player and the committee doesn't, that he has the right to go over their heads - I suspect in that situation he would need to get FSG (i.e. Mike Gordon) to OK it.
So I don't think the current situation is ideal, there's clearly a bit of tension there, but I don't think there's a major relationship problem.
I do think there is still money available to spend, but I think the current tension may be about the committee saying we should not panic-buy this week, and maybe set up a deal or two for January instead.
And finally, just a reminder that a lot of our spare cash is being spent on the Annie Road this year. I expect there to be more funds to spend next summer (it will be the first time in three years where we won't have a major capital project on the go), and this, perhaps, explains why we're OK letting some players run their contracts down. Realistically, given their injury issues, Keita and Ox wouldn't command much in the way of fees if they could be sold this month and Bobby isn't getting any younger so also doesn't have massive re-sale value. I expect next summer's window will see those players replaced by guys who are more viable long-term, and that there will be money available to do that. If there are opportunities to do those deals in January then some of them might be advanced (as we saw with Diaz in January this year).