The MIT licence permits you "to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software" provided that "the above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software".
You wish to modify and then distribute/sell copies of the modified version, which is permitted. The proviso is that you must include the copyright notice of the original rightholder(s), and the text of the MIT licence. Note that there is no requirement that this text be applicable to your customers; only that it must be included. You can, if you like, include it surrounded by (eg) a warning that it is included pursuant to an upstream licence condition but does not itself fully represent the licence of the purchased product.
It is perfectly OK for you to apply a proprietary licence to the code you are shipping, this being code you received under the MIT licence and then modified.