Now I'm nervous, though. Is there any safer way to deal with this situation other than just pulling the failed drive, let it rebuild, and then after it's finished, replace the predicted fail drive?
I'm not exactly sure what you were hoping for here. Replacing the failed drive in a RAID 5, then replacing the pred fail drive is the only way.
Should I be nervous about replacing a failed drive when I have another drive in the array with a predicted failure flag?
Yes. Particularly if you don't have a backup. If you don't have a backup, get one now, then worry about repair. A pred fail drive is technically still online, but it has reported to the controller that failure is imminent. Without knowing the extent of the damage or the cause of the pred fail, it's impossible to say how long it will be functional or how much damage has already occurred.