I am from the US, so I don't know if I fully understand your question. I will try to explain as best as I can. In the US, a CPA license is not the same as a degree.
We have to get a bachelors degree and typically a masters degree in order to sit for the CPA exam. A bachelors degree in the US typically takes four years, but honestly, there's no reason it shouldn't be completed in three. So, it sounds like your three year degree is probably the equivalent of our bachelors.
Most public accounting firms require you to have a CPA license in order to work for them. I don't know whether they care about your ACCA. I suspect it would be important to them if it truly is the equivalent of a CPA license.
Assuming they do want you to have a CPA license, every state has different requirements in order to earn it. The most strict requirements typically are 150 credit hours, of which 30 are from accounting classes, in order to sit for the exam. The exam comes in four parts that take about four hours each to complete.
All four parts must be passed within 18 months of passing the first part. Otherwise, you must retake previously passed parts. You also have to work directly under a CPA for two years in order to get your license.
This is only true in the most strict states, though; some have more lax requirements. Because of the class requirements in order to sit for the exam, most people major in accounting for their bachelors and then get an MBA in accounting or an MSA in either auditing or taxation. If I understand your information correctly, it sounds like you should get either both your ACCA and your LSBF or get just the LSBF and plan to sit for the CPA exam when you get to the US.
Let me know if you have any further questions about how it works in the US. I'll check back to see if you updated this question. Edit: Most public accounting firms require you to become a member of the AICPA, so you probably won't gain anything by getting you ACCA.
However, if you go to work for a corporation or other business instead, being a member of the AICPA typically does not matter. So, it really depends on what you want to do with your accounting degree / license. Most websites for accounting firms should have information regarding this in the employment section of their websites.
If you already have any firms in mind, I'd look into that. The major accounting firms in the US are Ernst and Young, KPMG, PricewaterhouseCoopers, and Delloitte and Touche. I would start there if I were you.
Those four also have a large international presence, so you may be able to find someone to speak to in person about this.
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