I know you like the truth.
The "lyrical" quality of the original seems like it was special.
I would strive to preserve it's charm by tracing and hand editing the vectors.
I'd charge them an extra fee to vectorize their logo and differentiate that from the screen art prep.
I'd give them a logo.eps and logo.ai in return for that fee.
That way they could have the hand hewn look and enjoy a consistent presentation in all of their communications.
It doesn't have to be a large fee and in the end it would provide a massive amount of value.
Said with love and respect.
From the android.