First time finding myself in a situation where I'm booking my stay partially using points, and partially with cash. How can I BRG the cash portion of the stay?

For example, June 1 - 4 award + June 4 - 6 cash.

Do I find total cost in cash of a room for June 1 - 6 and compare that with OTA, and then only if the average per night OTA cost is lower than my cash nights (June 4-6) will the BRG be approved?
What if my award and cash stay are booked into different room types (ie. award - standard room, cash - suite)?

Or is it simply not possible to BRG when you happen to make separate cash and award bookings? Talking to BRG CS is like talking to a wall... They insist that my "claim" is for the cash stay (which it is, since that's the reservation # on my BRG claim) but that's not allowed as the "claim" cannot be for a portion of my stay. Asking how to approach the situation with award + cash reservations, BRG agents simply ignore the question.