Why bother – apply for your FAC and for the price of a half decent air rifle you can get an utterly lovely .22 rimfire,
How big do you think his garden is?
This is like someone asking what bike to take round red trails and everyone telling them they need a Brooklyn or Karpiel.
An air gun at sub 12 will be quiet enough to work at their ideal ranges which won’t disturb neighbours or have the embarrassment of the local FEO laughing in your face when he realises you pissed £70 up against the wall to try and shoot rabbits in your garden with a gun utterly inappropriate for the job.
Rimfire is fine for clearing golf courses and working beyond 30odd yards. Provided the land is cleared and safe which unless he’s the Duke of Hampshire I doubt it will be.
Shotguns are shit because they are noisy, one shot and everything runs for the night. Again, unless you’re planning on moving far enough away to take the second shot you would be wasting your time. Just throw fireworks if that’s your plan.