Don't let the stereotype get to you. You are an accountant at this moment in time, you should embrace that and work to influence the stereotype. Even in FP&A I've struggled with stereotypes. I've spent the majority of my career convincing operations, marketing and management executives that being a finance/accounting person doesn't necessarily mean you sit in a corner and crunch numbers. That we can influence business, process and bottom line. That in fact we can be a profit center and not just a cost center. The point is, don't be frustrated by your stereotype, get in a position to change peoples minds and then do precisely that.
Regarding FP&A, I'm referring to FP&A in corporate but even then FP&A can mean a lot of different things. I've seen some that do strictly budgeting, variance analysis, month end close analysis, they usually report up to the controller (or VP of corporate finance). This type of FP&A is also a good alternative to the corporate accounting bridge I was talking about, if you can find a role like that.
The other end of the spectrum is an analytics driven FP&A department. Usually companies that employ this type of FP&A don't do a heavily amount of annual budgeting, but instead focus on quarterly or monthly forecasts. In these departments you will do plenty of forecasts and management reporting, but also a high degree of project analysis. For example, in one restaurant company I worked for, I led the monthly forecasting process (I had one analyst) that involved heavy discussion with the VP of operations, I led the marketing analysis function that involved marketing spend (was it efficient? Are they ever going to get an effective ROI from that stuff?), analyzed the use of contests for restaurant level staff (are they effective? Do they drive profit), analyzed a variety of higher end capital expenditure requests, etc. IMO, these are the best type of FP&A positions. They are usually pretty rare and highly coveted because you get to do a variety of different things while being in close touch with the business. But they take a high degree of business understanding and the ability to communicate sometimes highly complex financial topics to non finance people.
Usually FP&A positions will fall somewhere between the two opposite poles I've described above. You will have some degree of the month end stuff, some degree of budgeting/forecasting, and some projects along the way. Hope that helps!