I agree that you should have the skills. IV skills save lives. Been there. Even better, having the advanced skills that go with midlines, PICCS, all the many med ports and dialysis shunts would make a big difference on your floor and your career.
I'm concerned that the manager is sensitive to people trying to improve themselves. This suggests that the environment in which you work is likely to lack a basic level of trust. Ultimately that floor is not where you belong. Having said that, I'd look at what else you could learn that would not threaten the manager. Will she support you in taking ACLS or dialysis courses, or is that also considered "disloyal"? Can you use this job as a place to park until you've finished your next degree, or is that also not "important"? If the manager has a specific idea of "how you should become a better nurse", it may be worth riding it out, if the skills are useful elsewhere. If the manager feels that experience is more important than education, you're going to get bored and lose your enthusiasm.
Another side issue you mentioned is technically illegal in the US: working off the clock in another department for skill improvement, even if voluntary, places your employer in violation of wage-hour law. You can't volunteer for an employer. All hours have to be paid. If you were told that you had to work overtime and not get paid for it, you'd balk. I can tell you that whenever I did IV skills work at any hospital, it was on the clock. You deserve your compensation and respect for you as a professonal.
If you do want to volunteer and gain a lot of IV experience, I'd go find a rescue squad that will train you to an avanced level. Most of the time the IV skills are taught in conjunction with the hospitals that support the trauma program. I know I got a lot more experience with IVs as an EMT than I did in nursing school.
Best of luck to you.