So much to say here.
Your answers will vary based on the field and location that you're in. It will also depend on how long your previous stays at companies have been.
Firstly, I think you may be making some mistakes in your pretenses here:
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I would like to be able to have this company on my resume
The company on your resume means very little if you quit too soon. This again depends on industry and location, but a stint of less than 1 year is probably not very credible in most industries. In other words, the company on your resume is worthless if you're not considering staying there long (and are not a contract employee).
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I would like to be able to ... have references at this company
I would be genuinely shocked if you had a decent reference in less than a year. If you did, I'd assume that this is a "we fired OP, but we don't have any malice, so we'll at least give them a decent reference" type situation. In other words, a reference in this case would possibly be a negative.
Another aspect you mentioned, but did not dive into was that you work in a small field. It's possible that your short stint here may close some industry doors for you. Your coworkers (aka possible future managers, coworkers, interviewers) may remember you.
Lastly, your work history is also very important here. If you have little history or have made a few recent jumps, this will trigger some "job hopper" flags for future companies. You can explain why you're not a job hopper during interview... if you're not filtered out before you get a chance.
But of course short stints happen all the time.
Sometimes a company isn't a good fit. Sometimes life happens.
Some doors will likely close.
This job hop will likely become a small stain on your resume.
But usually both of those will get better after a long enough stay at a new company anyway. Being able to enter the company that you want to is probably very much worth these small demerits.
PS, if you have an emergency fund that you can live off of for a few months, this may be the time to use it. You can properly prepare for the job search, you won't have a notice period (meaning you can start right away) and you'll avoid closing any doors.
