You should disconnect all power to your sub-panel (turn off your main breaker and your generator). Be absolutely sure the power is off to avoid serious injury or worse. I'd suggest using a voltage sensor pen to verify the panel is cold.
You will need a ohm/continuity meter for this test.
Pop the breaker in question out of the panel, leaving the hot wire connected. Turn on the breaker and test continuity between the back of the breaker and the neutral bus. Since the breaker is tripping, you should see continuity/low resistance. If you do not, it's possible the short only exhibits under higher voltage but more likely, you have a bad breaker.
Assuming you do see continuity, remove neutral wires one at as time from the bus and recheck continuity. When you lose continuity, you just removed the neutral you're looking for.
That Federal Pacific panel has to go as they are dangerous. As far as concerns about your skill, you sound like that particular kind of newbie who is well capable of learning to do it all safely and well; however my hunch is you are still thinking too much, and need to read a little more. It's OK, we all start there.
Normally, just replacing a sub panel is a straightforward thing. Change panel, reattach wires, done. However, this is only part of a project with a much larger scope. You must contemplate (i.e. ask your permitting authority) whether you have crossed the legal threshold of a remodel. If you have, everything in-scope must be done to all current codes. Even ADA!
The 2-circuit requirement for kitchen outlets is not intended to mean "1 circuit for this wall, 1 circuit for the other wall". I'd encourage some more research but if it was me, I'd interleave the outlets, every other outlet on a different circuit. Also there's nothing wrong with more than 2 outlet circuits in a kitchen, the whole point is to prevent trips when the chef is madly at work, so the chef isn't hobbled with limitations like having to put the toaster here and the George Foreman over there, and avert ugly workarounds like extension cords draped across sinks or stoves, etc.
I see you plan to go 12 AWG wire for almost everything (that's what 20A breakers mean) - that's awesome. Feel free to kick the refrigerator and smoke detector up to 12AWG also - that way you don't have to buy any 14AWG wire. I don't own any! If you have some other reason to use a 15A breaker you are welcome to use that on a circuit wired in 12AWG.
You may want to run the water heater circuit in 10/2 or even 8/2. That will allow you to easily upgrade to an electric water heater in the future. Still use a 15A or 20A breaker because the outlet is still only good for 20A. There is a trick to fitting 8+ AWG wire on a 15-20A outlet, just ask.
I would go with a much larger panel. You have either -1 or 3 circuits left, and that's too little headroom for my comfort. Getting a larger panel is dirt cheap compared to the cost of replacing perfectly good breakers with duplex breakers (I call them double-stuff) merely to shoehorn everything in. Also, larger panels in combo-packs come with more breakers and that is far-and-away the cheapest way to buy breakers. Another reason to avoid duplex breakers is if you ever need AFCI, GFCI or whatever future thing comes out - those are much more expensive in duplex breakers because of the miniaturization required. Don't think you must use a 100A panel - you can use a larger panel (200A), you just can't use a smaller one (70A).
Remember each sub-panel must have its neutral bus bar separate and isolated from its ground bar. That means removing bond straps, magic green screws, neutral bar kits, whatever the panel requires to do that. You might consider a panel with a neutral and ground bar on each side of the panel. That's a convenient feature so wires don't have to cross over the panel.
Keep in mind how your house got a dangerous Federal Pacific panel. The last guy bought cheap. Feel free to research the good-better-best that each manufacturer offers, the price differential for "best" is quite small compared to the overall cost of a remodel. You may find better selection and better prices at a real electrical supply house.
There is nothing wrong with more ground rods. Go nuts. The key is that all the grounds are connected to each other by wire - and they are not connected to neutral anywhere except one place - the main service panel.
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Wait, what? Why would 240V 3-wire need the neutral bar??
Oh, right, this is a main panel where neutral and ground bars are supposed to be bonded, so someone took the lazy but allowed approach of landing all neutrals and grounds in the neutral bar, and not having a ground bar. So your neutral bar is stuffed.
The easiest way to free up space on the neutral bar is to get an accessory ground bar for that panel and move a bunch of grounds to it.
If the hots are all disconnected, why not disconnect the ground? Because some other circuit might be relying on that for ground. In older homes where everything was not grounded, it is common for grounds to be retrofit by running a ground from the previously ungrounded receptacle to a grounded site. While dodgy in the past, this was made officially legal and recommended in the 2014 Code.
So go to the other end of the wire, and look at the termination of that ground wire and the equipment it is grounding. If nothing else is using it as a ground, then feel free to detach and cap off both ends of the ground wire. In the panel, wrap it with tape (preferably green) so it's not at risk for hitting anything hot inside the panel.
Then take all 3 wires and push them into a corner where they'll be out of the way. I recommend not tearing them out, as they can be re-tasked for a 120V or 240V-only circuit for any future need. "Leaving them alone" is way cheaper than fishing new cable.