Hudson River didn't have dual engine flameout. The aircraft landed in normal law. Sully preferred Flap2 otherwise he could have landed in flap3 if he wanted. So Don't call that and mislead yourself.
I call it the Hudson checklist, not because he had a dual engine failure (which as you say he technically didn't) but that recommendation A-10-66 of the accident report recommended that manufacturers develop a checklist and procedure for a dual-engine failure occurring at a low altitude. Airbus developed the checklist in response to this IIRC.And its the big red button at the top of the eQRH and was on the 2nd last page page of the printed QRH. Hence my calling it the Hudson checklist.
It's a low level, I'm going to hit the ground / water soon and I don't really have time to troubleshoot checklist.
Fuel or no fuel is not the issue here. Whether Dual flameout or not is the issue.
For the higher altitude Dual Eng Fail, yes, Fuel or no fuel IS the issue. As you would know, the PRO ABN has two procedures. One for Fuel remaining and one for no Fuel Remaining. They are somewhat different (APU usage and relight phase) but are both different procedures to the EMER LANDING ALL ENG FAILURE abbreviated checklist - notably they both end up in Flap lever position 3 rather than Flap 2 which is the situation from the Hudson checklist. This I think is what the OP was trying to understand.
PS. For clarity I'm looking at A319, I assume the A320 Dual Eng failure ABN PRO is the same, I don't have an A320 QRH to hand.