It's not a dumb idea, it's almost certainly what's occurring, with the caveat that I don't think they actually want to shut the government down. Whenever you hear about an impending government shutdown, it is always a game of political chicken, trying to find out who will cave first, while simultaneously trying to preemptively sell the public that it's the other party's fault. Look at the messaging from the white House and congressional majority leadership. It's all "oooh the Democrats aren't willing to pass our super clean funding bill to keep the government open, they are unserious and willing to hold the American people hostage in order to continue mutilating babies". Meanwhile, Democrats state (accurately) that Republicans need Democrat votes to pass anything, therefore it is incumbent upon them to negotiate in good faith.
There's nothing more antithetical to Trumpism than good faith negotiation and compromise, so he's doing everything in his power to avoid that, lest it appear like he had to cave to Schumer, to include cancelling the meeting they had scheduled last week in favor of doing the meeting today, at the 11th hour, to further pressure the Dems into capitulating.
Now, it's important to note that, for all of the political brinkmanship on display, shutting down the government is, historically, far more damaging for the majority party than the minority, though the public tends to take a dim view of everyone involved in this sort of situation. Therefore, Dems have reason to stand fast and Republicans have an incentive to make concessions. This is in addition to the fact that Schumer got a lot of flak for instructing Dems to fund the government back in the spring, so he's also likely motivated to feign some backbone in this particular tete a tete.
I consider the occupation of certain cities to be mostly unrelated to the funding fight. In fact, it would have behooved Trump to not antagonize Dems leading up to this for the aforementioned reasons. With that being said, Trump doesn't do things according to what makes political sense, and, to your point, I can see a scenario where Trump is the only person at the negotiation table today who is totally ambivalent about whether a deal is struck or not. A local (Republican) representative was quoted with a statement to the effect of, "I'm not sure if the Dems have considered the fact that the Presidency is granted additional powers in the event of a budget related shutdown, maybe they should think about that".
So, you've got Dems at the table who are motivated to follow through with a shutdown unless they get certain carve outs. You've got congressional Reps at the table who are aware that the American public has historically always blamed the party in power when a shutdown occurs, and are thus motivated to make concessions, but quietly. And then you've got Trump et al, who gain additional emergency powers in the event of a shutdown, further diminishing the legitimacy of the other branches of government and increasing the consolidation of power within the executive branch. However, if the government doesn't shut down, Trump will likely spin this as yet another example of his brilliant negotiating ability (see also the Gaza peace plan released this week), even if there's a snowballs chance in hell that any concession appears in further funding bills the next time this occurs.
Idk if any of that holds any water in the face of evidence, but it's compelling speculation.
