"My idea is that, if it is legal to do so, you tell the receiving bank "this is a $10,000 rollover, but $7800 is coming from my old plan and $2200 is coming directly from me.""
That's what happens, but that's not what you tell the receiving custodian. As far as the Roth IRA custodian is concerned, the first transaction is a deposit of $7,800 as a Roth conversion. (The other $2,200 of the $10,000 is withheld for taxes, but the receiving custodian doesn't need to know anything about that.) Within 60 days of the original distribution the individual then makes a second deposit into the Roth IRA of $2,200 as an indirect Roth conversion to complete the Roth conversion of the entire $10,000 that was distributed from the originating account. The receiving custodian has no need to know any details other than these are deposits of Roth conversions. In fact, for an individual who is under age 59½, the $7,800 and the $2,200 are effectively two separate distributions which would be reported on separate Forms 1099-R, one for a $7,800 distribution with code 2 (assuming that the originating custodian has been told that the $7,800 is to be directly deposited into a Roth IRA) and the other for a $2,200 distribution with code 1 (the tax withholding). There is no limit on the frequency of Roth conversions, so there is no concern about these being considered to be separate distributions.
It's always a bit risky to tell the IRA custodian more than they need to know to perform the proper transaction.
