Basically what mla said – figure out where you want to be and then how to get there.
My employer is one that you could've had your job at, as far as title/duties, though it's a different work environment (small college as opposed to small mom & pop). We had an Assistant Controller position that was really Accounts Payable – maybe Accounts Payable Manager, if you're generous, or maybe just Accounts Payable Associate. But, it was called Assistant Controller. When the position came open and I'd been there long enough to know what the job really was, I got the title changed to something more appropriate to the duties…but, a paper-shoving (in this case mostly AP) job that's titled Assistant Controller is something I've definitely seen before. 🙂 The good news for you is that you've still got an impressive title on your resume. The bad news is some places will think you'd never take their Staff or Senior job, since they assume you're too highly compensated as Asst Controller to consider their Staff or Senior position.
My advice: if you want to stay in private accounting but move up to more challenging tasks, then do that. Figure out what level of duties you're doing, and look for something that's the next step up. Let's say your duties are mostly “Clerk” level; look for something that's a Staff Accountant or Senior Accountant level position. If your duties are already Staff or Senior Accountant level, then look for something that's Senior Accountant or Assistant Controller elsewhere. You might find an Assistant Controller job that's one step more challenging than the one you're at, and by moving to it, be able to get a pay and duties increase, even though the title stays the same on your resume; also less likely in that situation to have the “s/he's overqualified for my Staff Acct” issue.
If you want to do public, and/or want to work high-up in the bigger (especially publicly traded) companies, then get the public accounting experience. You'll likely take a pay cut to do so, since you don't have experience auditing, and auditing experience is really what they need in order to bring you in above entry-level. So, if you don't have a long-term plan using the public experience, there's no reason to take the paycut, but if it will be a step on the path to what you dream of and something that will repay the pay cut, then you might have to take it for now.