Your reasoning is incorrect.
In Street Name means that your securities are registered electronically in the name of the brokerage firm.
However, you are the actual beneficial owner and you own the rights associated with the security. They cannot be taken away from you because of litigation against your broker.
And FWIW, while the SIPC limit is indeed $...
Yes it is possible to do what you are asking. See following two transaction types at Fidelity:
Initiate an Exercise-and-Sell-to-Cover Transaction
Exercise your stock options to buy shares of your company stock, then
sell just enough of the company shares (at the same time) to cover the