I think no matter what you'll wind up with a timespan where the ham plate doesn't match the FCC records.
Logically, it seems that as soon as you the new call sign and the paper license arrives, send a copy of it to Lansing along the everything they want-have the MV-74 filled out as much as possible while waiting for the FCC and USPS.
The Sec of State I believe gives partial credit for vanity plates requested at times other than renewal. I would assume the same for ham plates. The SOS website has an 800 number that I believe rings into Lansing as it is the same number for both the Holland and Hudsonville branch offices.
Or do what I did when I got my ham plate: wait til your birthday is near. Just backtime everything (FCCland Sec of State) to figure out when you need to put wheels in motion. And with your birthday near a national holiday, I'd allow some extra time