Looking at the rules here the rules clarification for Double the Rent says (emphasis mine)
Play with a standard rent card to double the amount
The FAQ link to here it says
- Does the Double The Rent card count as a turn? Does it count as one of your three plays? Yes. An example would be a player lays a property
card, a rent card, and a double the rent card during their turn. This
would be all three of their card plays.
So based on that i would says that 'Double the rent' is playing with (ie simultaneously) with the rent card. The payment is 1 x 16 million lump sum because 'Double the Rent' and 'Rent' are played at exactly the same time.
I'm confused as to why asking for 16 million once over 8 million twice makes a difference as I've only played this once many years ago under duress and just checked the rules to answer this.