Depending on the time in may the south entrance may not be open yet. We stayed in Gardiner MT at the end of april start of May and much ot the park had just opened.
I was running 1982 f150 rear axle width. I felt like I was going over several times, I was younger and had the nerve to throttle through... I broke lots of stuff....
I did/do a hybrid. I drive easy at varied RPM (not speed that is not engine dependent) for the first 500 miles. Change the oil and then over the next 500 miles progressively bring up RPM and harder shifts. Change that oil at 1500 miles. You are ready to roll!!