Since you've only got a week or so, you have two choices: keep it taped and protected, or snip off the detached part and protect what's left.
If there's no good way to snip off the detached portion and you don't want to have a podiatrist do it (you need to do it ASAP to give it time to heal before your trip), just use a piece of medical tape and just tape it loosely but securely onto your toe. Not sure you need all the neosporin and whatnot unless there is evidence of infection (antibiotic ointments actually slow healing, so if there is no sign of infection, don't use it), but use the tape to just make sure you don't rip the nail off the rest of the way. That will slow you down a bit, and then you WILL have to do some wound care to make sure it doesn't get infected.
I've lost pretty much every single toenail on both feet numerous times during my marathon hey days. Take a nice photo to remember this by (cause you know you'll want to!), pick which of the icky choices above that you least hate, and go for it.
I haven't run a marathon in 3 years, yet both of my big toe nails are still only partially attached. Great fun!!