Instead of morphing one umbrella into the other one (using two masks and the reshape effect),
can't you just convert the Illustrator layers in your comp into shapes (Layer--> Create shapes from Vector Layer) and keyframe the paths you need?
Don't get locked into your existing designs. As Zulkifar already said, you need to animate stuff directly in AE.
I'm going to make a suggestion that you do your animation in Illustrator because you have started in Illustrator. You use the blend feature in Illustrator the create the transition between shapes and then you expand the appearance and release to layers, then import as a comp and release to layers.
Here's a zillion year old tutorial I did over at Creative Cow the lists the steps: Morphing with Illustrator
Just for fun I did a little test project because this is a good technique for a lot of things. This is what it looks like in Illustrator just before creating the blend.
This is after applying blend and expanding and releasing to layers:
Importing this as a comp in AE and sequencing layers gives me this:
When rendered gives this:
Just for fun here's the project file: Dropbox - umbrella animation.zip (CC 2014/ CC 2015)
you're awesome Rick! =)