So I ran the numbers from you:
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This is what the sky looks like at different times during Princess 2022 sunrise.
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These are the paces necessary to have that sky background given corral starting location. So for instance, someone starting in Corral D would need to run a 3:57 min/mile to see Astro Twlight, 9:35 min/mile to see Nautical Twilight, 15:12 min/mile to see Civil Twilight, and 20:12 min/mile to hit MK before sunrise.
So EPCOT in October is scheduled to open at 10am (9:30 for resort guests) and the Balloon ladies are projected at EPCOT at 9:16 and a 9:30am finish. So I don't know if you'll have much wiggle between EPCOT opening and being able to get a drink in there. So any drink would need to be handed to you by a guest on the Boardwalk perhaps (something they had discouraged more recently). That's more doable during the Marathon at Marathon Weekend. Not impossible, but tight given the current opening time. In year's past EPCOT opened at 9am, so then it might be more doable.