A word of caution first if I may. It is easy to spend a lot of money on a classic car engine and not really get much improvement. You are dealing with a 60-year-old design here and it will never go like a much less expensive modern car. Classic Minis are really for a 1960's 'retro' driving experience.
Now for the 'techie' bit. The key to better performance with any A-series engine is the cylinder head. It needs to be properly gas-flowed with the compression set to around 9.75:1. Then a better cam, a re-bore and new pistons, an HS4 carb, a 3-into-1 tubular exhaust manifold and a decent silencer system will make it go a bit better.
Build it very carefully and accurately and you may well get the bhp up from around 34 to 45+.
When increasing the power and torque you must ensure the gearbox is in top condition as more 'grunt' will soon show up any wear in the gearbox.
I think a budget of only c.£1000 is a bit low for what you want to achieve, even with you doing all the work yourself.
A classic car of any type is not entirely suitable for daily use and when my Grandson was 17 he had a fully and comprehensively restored Cooper 1275 MPI, but soon tired of it on a day-to-day basis and bought a 1.25 Fiesta Zetec. Of course, he is able to drive my cars, including my 1275 Cooper 'S' so he is not Mini-less.
Good luck with yours.