Rietumu is a reputable bank. Latvia does have banking secrecy, though it is unclear for how much longer, since the EU is more and more moving towards automatic exchange of information. It is unclear if this development will also include bank accounts held by offshore entities which in turn are EU persons.
I wouldn't recommend any bank. Bank recommendation is a complex procedure. It requires a complete understanding of your business and what banking services you need.
That said - FBME is a good bank.
There are a lot of banks that open accounts remotely for under $10,000. Virtually all banks in the Caribbean and Mauritius, and many Swiss banks allow remote account opening with less than $10,000 minimum deposit.
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