You can try fresh installation of Maximo 7613 + HSE and then point it to existing DB to run upgrade.
Running Integrity checker shall help eliminate all errors and shall be able to upgrade the base + HSE, as you said you don't care about control desk/service provider to be lost so overriding their class files shall not be an issue. You may still need to review for bean classes post upgrade. Also even after running IC, you may run into errors during upgrade which you can fix then and there.