They're both acceptable, but the first feels slightly more natural to me. A search brings up plenty of hits for both constructions.

When we want to refer to the past we use would rather (would sooner, would just as soon) + have + -ed form (perfect infinitive without to): She would rather have spent the money on a holiday. I’d rather have seen it at the cinema than on DVD.

I'd agree that they're both acceptable, but a general rule of thumb is to tuck the adverb in between the first two words of the three-word verb pattern (would have stayed).