British Audiologist wanting to work in the USA

Do you already have US citizenship? If not, then the most complicated aspect of what you want to do is going to be satisfying immigration rules. (Even if you don’t plan to actually immigrate, work permits are governed by a set of rules that fall under immigration laws.) For immigration matters, I strongly suggest checking on a forum dedicated to such topic. It’s just that it is beneficial to get input from a variety of people who have gone through the process, or are going through it. Even if someone here provides you with good information, I’d still check on a dedicated forum because you never know when there’s some caveat, or some recent rule change that affects your case, but was not mentioned by the person who helped you. The greater number of knowledgeable people reading your post and replying, the more chance you have to get the information you need.

For the doctorate, generally speaking, you need to come up with a research proposal that will overlap with the research interests of an established professor. If you can get a professor’s interest, you have more chance to get a fellowship.

If immigration issues are sorted out, I think hearing aid dispenser licensing/certificates are done on a state by state basis You’d need to pick a state and contact their licensing board for hearing aid dispensers.

Have you considered Canada? Our immigration system is not quite as turbulent as the American system is at the current time.
There are also certifying agencies that will provide educational equivalencies that accepted in the USA and Canada. Labour Mobility and NAFTA are not quite there yet. Health professions in Canada are regulated like the USA but by Provinces rather than States.

Plus you can get a PhD here as well.