Is there any way to advance in lab science positions without a graduate degree?

I investigated this at length when I was a recent undergraduate with a degree in chemistry from a prestigious school. The short answer is largely no—most lab science positions are on two separate tracks: one is for "lab technicians" who have either an associate's degree or a bachelor's degree. The other is for "lead/principle scientists" who have a Master's or, preferably, a PhD.

You might find some nontraditional opportunities at small or medium-sized companies, but largely if you want to advance in lab science (i.e. get a salary instead of a wage), you need an advanced degree. Without one, you're looking at being a "lab ass" which does not offer much in the way of advancement.

Ironically, big companies will hire PhDs for fields in which they have almost no experience. I talked at length with a guy who did his PhD thesis on semiconductor research, and ended up being hired to do adhesives research...

While it depends on which science and what experience, even a masters degree will often not be enough to "advance".

The real question is about what you want to do. "Advancing" with a Phd really means leaving the day to day work in the lab and moving into experiment design, justification and funding.

If you want to stay behind the bench you can do alright. Some day you will become the lead set of hands of a specific Phd scientist and they will be much more efficient if they have a long term relationship with you. To keep that relationship, they will have to figure out how to fund you well or you will run off to another Phd.

There's definitely a tier system for technician positions that depends on your experience and education. It may go something like: laboratory technician, research technician, senior research technician, research assistant I, research assistant II, research assistant III, research project coordinator. You can also get into clinical research, which may pay more (but no one in the field is doing it for the money).

Why not do a thesis Master's degree, at least? You get a better understanding of how research works, reading/researching studies, stats experience, and gain skills. You'd start at the level of ~research assistant with a Master's degree.

This is pretty much correct, with a big disclaimer that anything is possible. Every company is different. You could certainly work into a lab management role with just a BS if you're cut out for that kind of thing. You'd make decent money for sure.

Find a job, any job and get some experience, to start, even if the pay is crap. Once you have it and have a year of experience, start looking.

You have to take the first crappy job, otherwise, in the eyes of those doing the hiring you are simply under-educated, with no experience.