As you noted, this is discussed at my sibling thread (This particular thread is just on low margin requirements). To close out this question, pledging stock held in one custodian to another broker is basically impossible. * Custodian generally has zero incentive to even allow this. [1] * Even if the custodian allowed it, I can't find brokerages that even have a means to accept pledges from another broker. (most don't want to deal with it/have no system in place to account for it) * I've even dealt with a large ibank that holds both a loan department AND a brokerage. Even though the loan department is comfortable using the locked up equity as loan collateral, the bank's own brokerage still is unwilling to treat calls as covered. (comes down to the whole "this is too exotic and unless you are paying tens of thousands of dollars we aren't going to take time to do special research") [1] These sort of restrictions basically create effective barriers for all but the most wealthy (ex-)employees. Another example of this in the start-up world is the infamous 90 day rule to exercise employee stock (call) options when leaving that is even in effect pre-liquidity. No problem if value of options is trivial compared to your net worth; huge problem if it is nontrivial.
That would seem to solve your problem. Borrow against the shares and use the proceeds to cover your short-call margin.
Yup, this helps a lot and was a great recent find, but the interest rates can get so high it starts exceeding what I'm willing to pay. (Thus low margin requirements remain key to minimize interest fees)