To be real with you, probably nowhere.
Unless the people who set up the foundation for you can make some recommendations based on their network of banks that they work with. That's how foundations normally work. You go to a reputable service provider, explain your situation and wishes. They work out a plan and take banking options into account. They speak with their banking partners to get pre-approvals. Then the foundation is formed. The council of the foundation consists of majority (or exclusively) people that the bank already knows. Account number is issued and you wire funds in. Then you don't see the money again until the council says so.
Maybe some EMIs like Verifo, Zen, GuruPay, and Nexpay would take it, but there's a high probability that you'd be limited to EUR and SEPA only as they couldn't expose their correspondents.
Foundations are compliance headaches. They take a lot of time and effort for a bank to onboard and monitor. You've indicated 100,000 USD an approximate value of the foundation. How does a bank make money on that?
Second, you want to deposit USD, which is a very precious currency and USD relationships can be fragile. Uncle Sam won't think twice about yoinking USD correspondent accounts if a bank does something naughty. I'm not saying you're an international drug dealing, gun smuggling terrorist. But it only takes one bad client for a US based correspondent to kick the bank out.
Now, as for the jurisdictions you've specifically mentioned...
100,000 USD is not enough to attract banks in Switzerland and Liechtenstein even for a natural person. They want 10–100x that. Then add the complication of a foundation. A foundation set up, where? Nevis?
100,000 USD can get you a personal VIP/premier banking relationship account in Hong Kong and Singapore. But again, the problem is you're not just opening a personal account. Your application comes with all the complications of an offshore foundation. It's going to cost the bank dozens of hours to onboard and monitor/maintain. Relative to the revenue potential, you'll likely be rejected.
While banks in Mauritius have more modest requirements and are used to trusts and foundations, they prefer them to be local ones or at most Seychelles or BVI/Caymans. Nevis? What's that?
You might have some luck with banks like BONI (Nevis), Hermes Bank (St Lucia), Caye Bank (Belize), Turks and Caicos Banking Company, British Caribbean Bank (TCI), Cayman National, Belize Bank International, Proven (Caymans/St Lucia), Butterfield (Caymans/Bahamas/Bermuda), Bank of Asia (BVI), and MCB (Curacao).
But times have changed in the Caribbean. I wouldn't hold my breath.
You might have more luck shutting the Nevis foundation down and setting up another structure with a better service provider, who can help you make banking arrangements.
Finally, thank you very much for your answer.
Just to clarify the situation and where I am standing right now from my initial post:
I set up a Nevis foundation with the nominee as a councilor (not offshore nominee) to get the image of the real management. My target is only to get an eye off the jurisdiction of my residence to avoid any taxation. The funds are coming from another offshore company that is an operational company owned by the foundation and with the non-offshore nominee appointed to directorship. The funds never go through my personal ACC.
Understanding the problem with the banks, I decided that I would rather never store money in any bank for now, (given various cases of banks went bankrupt in all parts of the world as you may know plus possible blockages etc.) and would accumulate it on the brokerage account by investing in securities, so I put my funds to work and the risk to loose 100% of the money is minimal comparing to deposits. An additional perk is when I sell the securities I automatically get the valid SoW for those funds. I managed to open an investment account with a top-tier London firm just for the amount of 100,000 at the beginning. I was very lucky there.
So what I need is a 100% reliable solution to transfer money to/from the broker in one transaction. I plan to add around the same amount each year/two years. I will not keep money in the bank account.
As of now, I opened an account in the Belizean bank that you mentioned above, and Hong Kong EMI that seems to be unknown on this forum. After additional consideration and some failed transfer with the Belizean bank, I decided that I cannot use any of that for the amounts that I am not ready to loose/be blocked.
Mauritius Banks seems the best option for me right now, the only problem is that I have a Mauritius residence and opening account there can make it all useless. I am going to talk to some bankers in Mauritius regarding reporting to the local authorities, as the councilor/management is outside Mauritius. But I know very well their cultural background - in other words, it does not matter what they say, you should never believe any single word)
My currency is EUR and all the transfers should be with SWIFT in EUR. If any of the banks you mentioned in the Caribbean have a reliable and reputable correspondent for EUR it might help me a lot. Also, in case there is direct contact with Butterfield Bank, any email of the manager, etc. I would be very grateful if you send it to me in PM.
P.S. I understand how it intend to work with the trust companies, but I am not their client for now. The only trustees I would choose if I ever had enough money would be located in the IoM, Channel Islands, Cayman or Bahamas as the last resort. All other options like Seychelles, etc adds the huge risk of incompetence from the local peanut eaters that can simply transfer a million to the wrong account. Not saying about some more sophisticated things. The risk of fraud or dishonesty by the trustee is present in any jurisdiction, there are iconic precedents from Guernsey, as well as the IoM.
I hope this makes sense and any additional comments would be valuable.
That means I am unfamiliar with 95% of the new things as I read them here. So is the nature of a fast-moving world where lobbyists, politicians, and public servants keep moving the goalpost in order to rob us more easily. 
