The PP is correct, you could take a bus from DHS to the MK.
But unless you have a sentimental reason to want to go to the MK the first day (which it seems you might), I still think it sounds like a much more fun and relaxing first day to do DHS with your FP+, then the lovely boat ride to Epcot for a wonderful dinner, then strolling around WS before Illuminations to finish the night.
You don't say where you are staying, but even after Illuminations can be pleasant if you take your time leaving the park, stop in a few shops, then make your way to resort transportation.
Have fun whatever you decide. I'd say be flexible, depending on how your group is holding up, but you will want to make an ADR.