It depends on what you want to achieve. If you want to keep your mini relatively unmodified then stick with the A series engine. Going a Japanese conversion means alot of work and hacking to make everything fit and work.
For about the cost of... $3000-$6000? you can buy a whole 1275 engine, bore it out to say 1310, 1330cc and get it worked quite well to achieve about 70-80hp atw. That's plenty for a mini IMO. This includes gearbox rebuild depending on what state the engine and gearbox is in and how much work you do yourself. I am spending about this much I all up to build my 1330 excluding some work and tools.
With gearboxes, any 4 synchro box will do pretty well. Gearboxes don't break themselves, people break gearboxes. IMO remotes are best as rod changes do get sloppy with wear and are quite hard to fix without replacing the whole rod change unit. Remotes have better engine stability and are designed by John Cooper not some Leyland accountant
Don't forget about braking...
Speak to engine builders. Good ones usually have time to give free advice and will be able to give you a good idea what is involved.