Broke, 1 1/2 to 2 hours after high tide @ battery sounds about right. If you're looking at eldridge it also gives you the times of the current changes at the gate. You want the southwest ebb. You can probably time it to pass under the Throggs neck bridge 1/2 hour before the turn, if you are looking for slack. It's really only about two hour passage so you can be a few hours late for slack at the gate and still enjoy a nice ride down the river.
The tugs and barges also look for slack at the gate, so you are more likely to have big company with that timing. Not usually a problem. I've just slowed and let them pass, or waited until they come through. Monitor ch 9 and 16.
You want to leave roosevelt island to port as there's a rr bridge on the other side.
I would just check the local notice to mariners and make sure there are no " security zones" on the day of your passage. Sometimes they close the west side of the river due to a UN function.
The passage is about 15 miles, so missing the favorable current could add hours to your trip. I'd sit it out in either City Island or Manahasset bay or somewhere like that if I missed the turn, rather than make 2 kn. against the current. There's no diesel on City Island, that I'm aware of so if you needed fuel I'd use Manhasset bay.
The Max current can exceed 5 knots at time and in certain places It's unlikely that you'll be making max hull speed. against it. You'll get some chop that knocks your speed down. I wouldn't attempt it against max flood.
Never been through Hell's Gate but I would expect this kind of traffic on channel 13. Tugs and barges are not even allowed on Ch 9.