You may hit some diminishing returns there. Without the tarp (call it the “100 percent solution”) some water will always find its way in. The only way I know of to prevent it is to have a fully-hinged door like this one:
You’ve grown accustomed to the 100 percent solution, which keeps all the rain out, so it might take some time and experience to trust a 90 percent solution that allows some seepage. Your best bet is to renew the waterproofing in the hearth frame bricks (but not the hearth itself—ick! ugh! poison!) and if there is a gap at the lip of the hearth make sure you keep brushing ash into it as you sweep out your oven. The hearth and barrel are made of firebrick, which will absorb a lot of water before that water ever makes its way down into the sand.
Looking at your original post, it remind me that my spouse and I are still considering some kind of shelter over the oven. For example:
or this:
a third example:
And if you want some true outdoor living space, one of these:
Sharing these in the spirit of seeing what’s possible. They all beat a tarp, for sure.