Bedford Level Experiment [...] has the disadvantage that it shows that the Earth is flat.

I love this. As it happens, I live quite near Bedford and am terribly tempted to actually try it one day. (Edit Looking closer, turns out the Bedford Level is in Norfolk, not Bedfordshire, so a little less nearby than I thought.)

There are loads of fun ways of verifying that the Earth isn't flat. Some of these were easily available to the ancients - e.g. the shape of the shadow of the Earth on the Moon during a lunar eclipse (it's always a curve). Others are easier now than they used to be - e.g. the variations in the constellations you can see as you travel north-south (it's much easier to travel far enough to see this than it used to be).

Some, however, simply weren't available.

My favourite explanation for how we know for sure the Earth is round is that we've been up in to space and looked. You can even verify this yourself with a GoPro and a high-altitude balloon, which many hobbyists have done.