I'm sorry to hear it's so frustrating. I personally would wait at least another 3-6 months though before looking into other jobs. I hated my last job for the first 6-9 months, and then I grew to love it. In the end, I stayed for 3.5 years and only left because I wanted to switch industries. I just think 3 months is a short time span and lots of negative emotions can be due to transition/being new.
Looking back on it, one of the reasons that things got better was because I began to feel valued and good at my job, when I hadn't at first. I imagine that going from the golden child to a cog in the machine would be hard to adjust to - and that validation would be missing. You're reading a lot of signs that you're not important, that you're not needed, and of course that sucks. But if you were the golden child before, I have a feeling that you will start heading in that direction sooner rather than later. You will soon better understand how to navigate the technical staff bullshit, you'll prove your worth, get clout, etc.
The other reason is that, ignoring the workplace environment part, your setup sounds great. It would be a shame to throw a structurally great job - negligible commute, good hours, flexibility, and good pay - because of a rough start. If you turn the corner at this place, you might want to be there for years and years. The alternatives you're imagining probably look exactly like your current job but the environment doesn't suck - in reality, you might be able to find a place with a better environment, sure, but with a longer commute, longer and stricter hours, etc.
Those are the reasons I would wait. I wouldn't worry as much about trying to get a good reference or to hold the job for some minimum amount of time. You have good references from your golden child company presumably, and you can just tell potential employers that your last company wasn't a good fit. Anytime someone makes an unhappy exit, the expectation is that the current employer will not be used as a reference (otherwise, the prospective employer would be outing you to your current employer...).