Hi Klippie,
I believe that this module used the old #355 batteries originally. From what I have learned from retroleds some of the old Armitrons used the same module as the old Saturn watches did. You can use the Pulsar style spacers with the #357 batteries also I believe. It is a bit larger than the #386 and might or should last a bit longer.
I am sure that more knowledgeable folks will be able to help with this.