Based on what your goals are, your best bet may be to check into a regular experimental psychological master program. I would recommend applying to a program like Montana State University:
It is a fully funded terminal master program, which will prepare you for a PhD program. It is not necessarily heavy in statistics, but they have some excellent cognitive researchers (see Dr. Hutchison or Dr. Meade), as well as health researchers (see Dr. Skewes and Dr. John-Henderson). They also have some researchers who focus on neuro-psychology (Dr. Babcock) and eating disorders (Dr. Lynch). What this means is that you will get a stipend to teach, conduct research, and could likely focus on the specific disciplines and topics (I know a few graduate students who learned under Dr. Hutchison, who also collaborated in other labs).
You will get plenty of applied experience designing your own experiments and conducting statistical analyses.
Psychometrics is somewhat different than your typical research. It tends to be more evaluative, and more based on understanding how items function on a test, as opposed to conducting experimental manipulations of human participants. In fact, common statistics in psychology are hardly ever used in psychometrics. Psychometricians learn measurement theory, although there is a lot of overlap (again, it is just applied somewhat differently).
Sadly I am unsure you will find much in the way of psychometric programs who only teach you at the masters level. It is more likely to find access to those courses in programs at larger institutions where you will be trained for a PhD (though again, some PhD programs have you earn a Masters first).
Honestly, at least as of now, psychometricians are in high demand, so even if you took 2-3 courses in it and taught yourself, you could probably get a job in the field. I know someone who worked at a local college doing program evaluation and item assessment. They were working on a masters but left for a job once they sat in on a few courses which discussed Item Response Theory. These fields tend to be less linear, and it is largely about what you have demonstrated experience with, unless you want to take an avenue which develops your skills (generally a PhD program).
It sounds to me like there are multiple reasons which are pulling you toward taking a Masters program first. This is becoming far more common these days. However, with a terminal masters you do not usually get to specialize too much, and it is really what you make out of your time, which is why I think a program like the one at MSU could be a good fit.
Disclaimer: I do not work at MSU nor do I have any direct affiliation with the faculty mentioned, just individuals I know of / whose work I read / whose students I know.