Try putting the gear selector in the neutral position . park neutral safety switch . Wouldn't need reprogramming , if it did you wouldn't be able to do it . Dealer only . Your best bet mite be to take your vehicle to a qualified repair shop any way . Viewing wiring diagrams for the starter circuit an using a DMM - digital multi-meter to test B+ power & ground's is how to find a fix. Do you know what a wiring diagram is ? How automotive electrical circuit's work ? How to use a DMM. Electric Testing Techniques You Need to Know
Free wiring diagrams here http://www.bbbind.com/free_tsb.html
Enter vehicle info. year , make , model an engine size . Under system click engine then under subsystem click starting . Click the search button then the blue link . Looking at the diagram i see the ignition switch supplying B+ voltage to the starter cut relay ! Do you know what a relay is ? If the B+ can't make it through the coil side of the relay to the park neutral switch to ground the relay will not energize an close the contacts an send B+ voltage to the starter solenoid . You can test at the relay for B+ control from the ignition switch . Video's on youtube on checking relay circuit's
Not sure where you are having the problem, but if you removed the bolts on one of your motor mounts and use a floor jack to move the motor around a little that may give you the room you need, Just be careful not to go to far and brake something like a wire etc, good luck.