Do I HAVE to have a PhD to get a post-doc position? I suppose this is probably largely lab dependent, but I'll go ahead and explain anyway.

I graduated last year with a bachelor's degree in engineering, and landed a job as a Research Assistant (negotiated up to Research Data Analyst). I was hired at the same time as a Post-doc fellow, doing very similar work together. I have done well, and for various unfortunate reasons my colleague has not done so well. Long story short, he is quitting from frustration. 

In contrast, the PIs are very vocal in their appreciation for my work, allow me to take place in interviewing candidates for his replacement, and allow me to take on more responsibility and make sure I feel like a valued  member of the team. Add on the fact that I have learned a tremendous amount in the last year here, and I am feeling very confident in my abilities (after all, I succeeded with a bachelor's where a PhD struggled)

I have been entertaining the idea of seeking new employment for financial reasons. They do not pay me what I believe I deserve. They might, if I showed them an offer from another institution though, and many of the job postings I've seen would suit me extremely well - I have all of the required and preferred skills. I would need very little training and love the work.

The problem is a good deal of the job postings specifically ask for a post-doc, because the typical bachelor's degree would be unqualified for this type of work (never published, can't read papers well, and doesn't have the computer skills to survive), although a couple postings did mention they would hire based on skills.

So in your experience, if a lab is looking for a research assistant type of position, and are hiring post-docs for it, are they specifically looking for a post-doc, or would they hire a research minded 'employee' with a lesser degree?

Am I going to have to go back to school for 4 more years to apply for jobs I'm already qualified for?
postdoc without doc cannot exist... You might be hired as something else, another description, same job, but not officially as a post-doc, by definition.

As commented, the funding might just not be there for such different position. I'm sure that, if I'm reviewing spending information from a grant, I wouldn't agree with that switch, unless it is significantly cheaper. Further, most of the sponsoring agencies I know, have a some sort of guidelines for paying people, and it would be difficult to justify someone without a phd receiving postdoc level money...

Your question sounds like "Do I have to go to medicine school to be a surgeon" or "Do I have to pass the bar to practice law". Doesn't matter that you think you are qualified, officially, you are not.