Modifier

Here is my line of thought.

Two main points :

1.Peering is a VERBing modifier here and not a continuous tense Verb
2.I suppose the reason for your confusion is the past participle looked.

Now,lets understand what the sentence is trying to say.

When she was looking at the cat in the past, during that instant she was peering/looking out of the window.
Here the tense of Peering doesn't imply its in present tense and rather implies the modification.

Now lets understand the implication of VERBed here(past participle as meant)

Peered out of the window,she looked at her cat.

This sentence clearly becomes non-sensical.

Peered seems to modify the subject of the coming clause and that is SHE.
Does it make sense with the subject -Yes.
It seems as it acts as a VERB and not a VERBed modifier.

Also,

The sequence of verb is not clear.(peered and looked)

Hope it helps.

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