The amount of snow that will be present on Whitney this spring is going to be a lot more than a “bit of snow.” You’ll definitely need crampons and an ice ax to climb up Mt. Whitney in May. The snow consistency will be much different on Whitney in May compared to Taos in the winter. If you have experience using crampons and an ice ax, keep the date. Otherwise decline the permit and look at getting a wilderness permit for entry from New Army Pass/Shepherd Pass/Kearsarge Pass with a Trail Crest exist for around late August through September. Those permits are easier to obtain (the remaining 40% of the quota open up 14 days prior to the day of departure) and you only deal with the Mt. Whitney Trail crowds on the way out.