If you have to miss, you're going to have to do what I've had to do a few times when I couldn't make the beginning of the draft. And that's to edit your personal pre-draft player values and run auto-draft for a few rounds before you can make it. By doing this, you can direct the computer to target certain players to a point and not target others you don't want. It works pretty spectacularly if you put the time into it.
If there is a time that works perfectly for everyone but you, that's probably going to be your option. Asking 8 other people to inconvenience their weekend isn't really a solution to the problem.
Additionally, if none of the times work, maybe Stallion would be open to getting up in the early to draft on a Sunday but that'd still have to both work for most everyone else and that would push the draft at the earliest to be from ~3pm-5pm EST on a Sunday.