When applying for the Work Authorization, you will have to select which elegibility applies to you. You are going to have two options:
K1 - This will be only valid for the duration of your K1 visa (max 90 days, i.e. the permit would expire at the end of the 90 days after entrance with K1 visa) - in practice, I'm not sure how useful this is at the moment, as it is possible, if not more likely the permit would not be ready on timely manner, considering current processing times.
Jointly with AOS. The duration of this one will be longer, and probably work for the time you wait for the Green Card;
Processing times for EAD is heavily dependent on where you are applying for it. If you look into the timelines at VJ currently, some people are getting the work permit through AOS in roughly 30 days, others in a couple of months, and others have been sitting for quite some time.
To be fair with you, and this is not advice, I see no point in applying it through K1 currently. AOS seems to be the only viable option. But I may be wrong here. To be fair, haven't heard of anyone applying for the K1 EAD in the last few years. Maybe someone else have head.