Is it wrong to turn down a title "promotion" and more responsibility if my employer won't increase my pay?

I was supposed to receive a promotion at the beginning of 2018. Didn't happen cause I was on maternity leave and I guess the company didn't feel I deserved a promotion although my boss and coworkers did. Whatever, maybe next year, I thought.

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This year in February I asked my boss why I hadn't received a raise or promotion and she was under the impression that I had. We went to HR to get an explanation and my HR rep just told us that budget cuts wouldn't allow for a promotion at this time even though I had checked all the boxes and could not possibly do anything more to receive a promotion. HR rep also said that she personally had approved my promotion, and it was someone else within the organization that came back and denied it.

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So that was unfortunate and I've been looking for another job. However now my boss and the Principal Investigator in charge of all the projects I'm on (he isn't my boss or my boss's boss, which is an organizational issue time and time again. Sidenote, when he found out I didn't receive a promotion he was pissed off and marched down to HR and laid into them, but nothing has changed still) want me to be a certified trainer. This GREATLY opens up our staff since a CT has to escort employees of less than a year basically anywhere in our lab. It doesn't come with any additional pay though and the only benefit to me would be to have it on my resume.

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I know the advice here is almost always to find another job but I'm really reluctant to do that since I'm trying to get pregnant and the US does not offer FMLA for employees that have been with a company less than a year. Furthermore the field I'm in doesn't treat pregnant employees well at all. Moving companies and then not being eligible for FMLA for my last child is practically not an option.

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So that leaves me with either accepting the CT title with no additional pay, or going to my project manager/my boss and telling them I'm not going to do it without more pay. I know my job is safe cause like I said, new employees don't get to do any work for almost a year because our training takes so long and we have a staff of 6 expected to cover 24/7 shifts. It really seems shady that the PI and my boss are able to make me a CT (long approval process, many people are expected to contribute their opinions on me and my work ethic) but they can't manage to get me a promotion after working here for 3.5 years. So yeah, they reeeeally want me to be a certified trainer but I'm still a level I? Come on.

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tl;dr: boss and coworkers want me to be a trainer (more responsibility, more hours) but company won't budge on giving me a promotion.

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