Is it permissible to deposit a company's dormant funds into an Islamic bank, such as Al Baraka Bank, in an account with a variable return, knowing that the company deals with interest-based banks through current accounts that do not accept interest? And can the variable return be used to pay taxes and customs duties, or for all purposes?

It is forbidden to deposit money in usurious branches unless it is for safekeeping in a current account only. It is not permissible to deposit it in a savings account, even if the interest is disposed of, because it constitutes a usurious loan. The Permanent Committee has stipulated the prohibition of depositing money in usurious banks, whether with or without interest, unless there is a fear of its loss and no other way to preserve it exists except by depositing it without interest, in which case it would be committing the lesser of two prohibitions.

As for investing money, it is permissible in banks whose transactions comply with Sharia. The bank mentioned in the question invests in usurious government bonds, so it is not permissible to invest money in it. Rather, one should limit oneself to a current deposit when needed, and it is preferable to deal with a purer financial institution.