Can a Visa DR card be attached to the assets held in custody that draws on any cash balances first and then taps into the collateral’s equity next?

This depends on the bank, usually they work like a line of credit that can be drawn down and transferred to an operational bank account. The swiss banks/private banks tend not to have debit card access but we can set you up with a structure – swiss/private banking + international bank to achieve what you described.