Taking possession isn't an issue, as long as the person still is within the 60-day window?

"for Broker 2 to re-code the account"

No.

I agree that it should all be treated as Basis and conversion to Roth. Yes, the 1099-R(s) will be issued as early distributions, but as long as you have the due diligence to show what transpired and they are in the 60-day window, it should withstand scrutiny as a Rollover of Basis, a nontaxable conversion. What you want to do is check for withholding, of course. That will need to be added back.

First 1099-R: complete distribution of Roth IRA, exception applies to early distribution, since it was "rolled" the same as if it was put into an employer plan account, as far as the Roth IRA disbursement agent knows.

Second 1099-R: Conversion of Basis (might have some taxable earnings while they've need trying to deal with this mess) to Roth. It should be done Direct, to avoid withholding.