Yes, Mikegtr has it right, this banjo was designed for nylon (actually gut) strings. This is why the neck has bowed forwards. I'd guess it dates from the late 1800s, but the Banjo Hangout guys will probably be able to say more.
Even if you "straighten" the neck as you propose, it will probably continue to move if you keep steel strings on it. Plus those tuning pegs don't work at all well with steel strings.
So, a nylon or Nylgut set, from Aquila or Labella most like, is the way to go.
Fixing the bowed neck
Adding wood to fill in the bow is really not the right way to go. It changes the geometry of the neck, which will make it feel odd to play, and won't deal with further bowing. Plus it will be really hard to make the addition look like it should be there, though you might be able to work out some way to make it look right.
There are two possible fixes, both of which require the neck to be straightened first after removing the fingerboard. To straighten the neck, put spacers under the two ends and clamp the middle down to your bench. Heat the neck (using a heat gun or hair dryer) until the whole thing is hand hot, being careful to go slowly so as not to damage the finish. Leave clamped up for a few days to see how it has reacted. Repeat with different sized spacers until it is flat.
Then you have three choices:
1. Reinforcement, either with a steel strip/T section/square tube or, more sensible, a carbon fibre strip or square tube. Rout a channel down the middle, epoxy in the reinforcement, plane the surface flat, reattach the fingerboard. There is a lot of guesswork here - you want the neck to flex slightly, enough so that if you held down a string at the body and just next to the nut, there would be around a 0.1-0.2 mm gap between the middle of the string and the fingerboard. This is known as relief. A completely flat fingerboard makes clean notes difficult, because the string moves up and down most at the middle of its arc and can hit the fingerboard. Too much relief means you have to press the string down too far. At a guess something like 6mm x 9mm carbon fibre tube, inlaid so the 9mm is depth and 6mm width, might be right, but this is only a guess.
2. Add a truss rod, again routing a slot in the neck. A two-way truss rod would be best, as it allows you to bow the neck forwards and backwards, but you might not have enough neck depth for that. You'd have to rout deeper than the truss rod so you can glue a fillet of wood over the top, as that fretboard is too shallow to take upward pressure from the truss rod. If you decide to take this route I can write more. The neck will be much heavier, which changes the balance of the instrument.
3. Glue on a thicker fingerboard - the easiest way is to add, say, a 3mm board to the neck and then the existing fingerboard on top of that. This might be enough, but again it's guesswork. I'd choose mahogany for the addition, as it's lighter and also less stiff than ebony. If I had the neck in hand I might guess lighter, maybe a 2mm or 1.5 mm addition, but that would be from experience based on how the neck flexed by hand. So again, guesswork here.
Removing the fingerboard
As others have said, you simply heat it with an iron and ease it off gently. This fingerboard is almost certainly glued with Hot Hide Glue, and heat alone won't soften it, you need moisture. However, if you use a damp cloth between the iron and the wood, it's very likely that the fingerboard will curl up along its length - not good!
I would heat the board dry, with a cloth between the iron and the wood. Removing it from the hoop will make it easier. You want the wood to feel hot to the hand, but not hot enough that you can't hold a finger to it for a couple of seconds. Once the section you are working on is that hot, work thin blades under it from either side (start at one end, and I'd suggest the body end is best). The Works will sell you a pack of four or five artist's palette knives for around £5, and these work well. Dip a blade in water, work it gently under as far as you can, repeat the other side. Keep going on alternate sides until the whole fingerboard is detached. Never lever it up! Work horizontally.
Runout is the gotcha. Your palette knives might start to dive into the neck along a grain line, or up into the fingerboard. Watch out for this and correct immediately. If you go slowly and take your time, it should come off in one piece.
Also, watch out for the "pip" which holds the 5th string above the fingerboard near its peg. If this is bone or ivory, heat won't hurt it. But it might be celluloid, in which case it will melt! Remove it if you can, or be prepared to make another and fit it once the fingerboard is back on.
The inlays are probably laid into some kind of back mastic. That will soften, but should harden up again. Be careful lifting your cloth when you remove the iron, in case the mastic sticks to the cloth and pulls the inlay out. If any rise up, flatten them again before it cools.
The side dots look to me like later additions, probably plastic rod glued in with CA glue. These should stay in place OK.
Finally, once the board is off it might well have distorted. Remove glue from the underside, iron it flat, and then clamp it between two boards to keep it flat.
Good luck! This won't be a quick job to do right.
But adding 2mm to the fretboard only adds 2 mm to the bridge height (assuming it's right at the moment, which we can't know), that should be no problem.
