There is an option to have the shopping cart on the right, but the search box has to be activated:
Reactivate the search box module.
Go to Modules - Positions. There, look for displayTop.
Move the search box module to the first position, and the cart one to the second.
Now, go to /themes/default-bootstrap/css/modules/blockcart/blockcart.css and on #header .shopping_cart{ add float: right;
Then, go to /themes/default-bootstrap/css/modules/blocksearch/blocksearch.css and on #search_block_top{ add display: none;
This makes the search box invisible, it keeps been there, but at least you can't see it and the shopping cart stays on his position.