most people don't send out engagement announcements. They tell their family, then their friends, as they choose the bridal party, and announce in the newspaper. The bridal shower is for the bride's friends. Both families contribute to the guest list and the wedding invitations go out. The RSVP's confirm the guest list.
If your mother=in=law to be insists on engagement announcements, gather the guest list from your side of the wedding party. Give your family two weeks notice, as it is "family" tradition with his side.
Set aside a day with her for tea, and bring along your guest list. You can then compare guest lists, don't hand yours over to her, just suggest you compare guest lists.
She will want to know the design of the engagment announcements. Because it is a new tradition to you, (tell her it is, she knows it is a new tradition to you) ask if she will accompany you for input and help you choose this most important tradition. Don't slight her here, for this is her one request. Go in and browse together and then choose.
Soon, get your mother=in=law cooing over her engagement announcements, don't frown or laugh at her tradition, smile sweetly, and be happy your family allowed her this tradition, as you will soon be sharing grandchildtren.
Maybe bring some bakery things for the tea and the proposed shopping for the engagement announcements.
Etiquette is important to her, and find out her family traditions.