It will probably be cheaper if you do the regular registration but may not be easier.
First you will need to fill out a late stallion report and that will be $25 and I don't know if the person who bred the mare has to do it or if it can be the current stallion's owner. If she is over 3 years of age then it will be $60 on registration fees. You will also need a stallion service certificate signed by both stallion and mare owners and again I don't know if you will have to have actual signatures from the owners at the time of service. I'm pretty sure you do. Also the stallion and dam must be brought pernament if they aren't already.
So your looking at maybe $100 especially if your membership is already paid for, whereas if you just hardship her in using her AMHA papers your looking at paying $260 if she is 3 years of age.