Can non-US institutions sponsor US crypto businesses legally?

No. All in all, this is a BS claim.

As I have already cited in my other answers, please do not be fooled or conned by someone claiming to provide you with such an arrangement. Almost every licensed bank knows of this and many licensed non-banking financial institutions are also aware that their license cover does not extend out to the United States.

I also wrote another similar answer: Faisal Khan's answer to Are licensed financial institutions outside the US (like banks, MSBs, etc.) able to provide money transmitter license coverage inside the United States for my remittance project?

No foreign licensed bank or non-banking financial institution can provide money transmission cover inside the United States until and unless they are duly registered and licensed within the United States.

This is especially true for a lot of banks in various offshore territories, former Soviet Republics or countries not particularly known for their banking that will claim their banking license extends into the United States and your crypto exchange/coin/wallet/trading project is covered by it.

It is not.

If they insist, then please ask them to provide a legal opinion by a well-known US payments and banking law firm that says so, or perhaps they have an opinion or an answer from one of the US state regulator?

Any agent or delegate claim, if offered, is only valid in the state, jurisdiction or country they are licensed for. For example, if a bank from US Virgin Islands (a US territory) or a Bank from Puerto Rico (also a US territory) claims to offer you blanket coverage all across the United States - via a delegate agent relationship, this is not correct. You can seek a legal opinion on it and the answer will be the same, the authorized delegate privilege extended by the banks there, are not applicable/acceptable in other US States / US territories and especially not accepted in the state of New York, where the BitLicense is required. For a Puerto Rico bank to be able to offer money transmission license coverage in the 49 states of the United States, it needs to have a FDIC charter. Most of the so-called banks in Puerto Rico are IFE Banks - International Financial Entities (IFEs) that are licensed and regulated by the Office of the Commissioner of Financial Institutions. They are very much limited in scope (check the OCFI’s website on the allowed activities, etc.)

My standard disclaimer of IANAL / IMHO / YMMV applies.