Flattening - how I'd do it (a variation on 'scary sharp'):
The idea is to use a narrow strip of abrasive sheet, and chock up the sole so that said sheet only abrades the sticking out bump in the plane and doesn't do the whole sole. The "chocks" are electrical tape, encouraged to slide on the glass by lubricant. They stop the plane see-sawing.
Start by establishing exactly what the shape is - where is the convexity and how bad is it?. Once you know, stick down a strip of coarse wet+dry to a large piece of plate glass (my plate glass is about eighteen inches square). The strip only needs to be about half the width of the plane. Fix it down with glue from a new can of SprayMount (stand the can in really hot but not boiling water before spraying to get a finer spray, and spray the back of the wet+dry, NOT the glass!). Smooth it down immediately and solidly with a wallpaper seam roller. Give it time to dry off a bit before use.
On the plane itself, apply small squares of electrical insulating tape to the corners of the sole. Cut them with scissors or a knife so you don't wrinkle the tape. Build up two or three layers, or more, as necessary, on the corners, so that they support the sole high enough to cut in the middle area but not at the edges, and that it doesn't rock on the glass plate - may not be the same quantity in each corner, if the plane is convex because it's distorted, all bets are off as to how much you'll need.
Sight under the plane against a strong light to see that the bump is just lifted off the glass. Check it doesn't wobble, pack whichever corner needs it with tape. When it's solid, add two more thicknesses to each corner to lift it clear by roughly the thickness of the wet+dry.
Pick something suitable to lubricate the tape sliding over the glass - talc would probably do, or Liberon wax, but I'd proably stick with soapy water.
Blue the sole (big, permanent felt-tip pen), before starting, then remove individual pieces of tape in sets (one from each corner) until it starts to cut the high point. Work it gently until you have a flattened area and the wet+dry stops cutting. If that's too small (i.e. it's really uneven), take off one patch from each corner and repeat.
Use water with (literally) only one drop of washing up liquid per jug-full as cutting lubricant. Photographic wetting agent is even better if you can find any nowadays.
Find a strong magnet before you start, and put it in two clean polythene bags. As the grinding crud builds up, frequently remove it by passing the magnet low over the abrasive. When the outer bag has a pronounced 'beard', remove it from the magnet by turning it inside out, wash it off and you're good to go again.
This should give you a fairly large, dead flat area on the sole where the bump stuck out. when you think it's big enough so that you can balance the plane on it without rocking it (for the next step), stop. If it stops cutting but the area is too small, take off another patch of tape all round to lower the plane by that thickness and expand the area.
At that point you can remove all the sticky tape, and the wet+dry, clean the glass with meths, stick a fresh full sheet down and gently have at it again to get the whole surface down flat.
I know what people say about having everything in place on the plane to avoid it distorting, but if you look at the geometry of a Bailey plane, the frog causes little bowing effect unless something else is wrong with the machining to start with (a bigger problem). I've done mine with everything stripped off the main castings. That also lets you use one the threaded holes to fix on an improvised winding stick of some sort (if you want to), which should warn you if you're rocking the plane without realising it. If you're really concerned, I'd do the main flattening with a bare casting, as it's much easier, then reassemble it and finish it off once it's flat enough not to wobble, but honestly it's not a shoulder plane so the geometry and forces being applied are quite different.
It will take an evening or so to do the first bit, probably (on the kitchen table, in the warm, with a radio play on in the background), because you have to be careful. You also need a lot of newspaper because it's messy, and the damp filings will rust like crazy if left around.
When the sole is flat, finish off by wiping quickly dry then squirting it liberally with WD40, so it won't flash-rust (WD40 drives off water). I'd also store the bare sole on a radiator in this weather, so it's immediately in very arid air.
I haven't done this with a totally convex sole, but i have flattened several planes very nicely with a glass plate and wet+dry. They are sufficiently flat for stiction to be a nuisance when plaining some hardwoods (sometimes!).
I have also solved a similar problem with a handheld power plane: The two sections of the sole weren't in parallel planes - the front adjustable part had been cast badly, then machined out of true - tilted in two axes. In that case I kept the reference sole of the plane clear of the strip of wet+dry by strips of insulating tape. It worked very well, and salvaged an otherwise totally useless tool.
It occurs to me that you might help the alignment as you work the plane to and fro by fixing a batten to the glass to run the edge along. Do this with d/s tape, before everything gets wet, and weight it down for a short while so it doesn't immediately come loose once the water gets near it.
But then I don't do this stuff every day - only when I need to!
E.
PS: Forgot to say - float the plane over the abrasive, don't push down (put weight) on it. If you do push down, you roll the particles of abrasive over /bed them into the paper more, and they just blunt off faster and don't cut as well. it also clogs faster and more permanently. Push down just enough to feel the resistance across the paper, no more.