The process for determining whether a driver is willing to transport a pet is asinine. Just add something to the app so the passenger can indicate that they have a pet, whether it is in a carrier, its approximate size, etc. and let the driver decide whether to take the job.
These services are already borderline questionable given all of the problems and the increased prices now that they aren't trying to put taxis out of business. The moment one must deal with someone at the company to beg for a refund or deal with any of the other problems the equation becomes pretty one-sided.