I feel a bit of devil’s advocacy coming on…
Whilst I can understand where you are coming from, I think we need to start by looking at the wider picture to make sure that we don’t end up with a solution looking for a problem.
Rail replacement bus services, by their very nature, put passengers to inconvenience. Simply having to make an additional change or changes, and taking longer to get to one’s destination is an inconvenience in itself. This can be made worse if people are carrying luggage, and there are certainly some rail replacement buses that I’ve seen, for example Optare Solos being used on Melksham replacement services, where a family of four coming back from their jolly hols with a suitcase or two each, would cause a problem on that particular bus, especially if it happened to coincide with a wheelchair user needing to occupy the disabled space.
Whilst I am aware that there are people on this forum who travel with their dogs, my simple observation on trains suggests to me that the proportion of canines as opposed to people on the average train is infinitesimally small. Indeed, I would hazard a guess that there are more cycles on trains than there are dogs and, as already been pointed out, these aren’t allowed either.
Given the size of the problem, therefore, I wonder whether it is worth making a fuss about. Graham might be right in saying that there are more dogs on trains at weekends because there is more leisure travel, but leisure travel is not always date specific. By this I mean that, as a proportion of the total, not very much leisure travel has to be done on that particular day (eg attending a planned event); going to spend a day with Auntie Flo in Bognor, or going to Stratford on Avon just to have a look around, are things that can be done at weekends where the full train service is running.
Finally, I would expect that the bus drivers themselves would have ultimate responsibility to decide who or what they allow on their bus. For example, a driver may refuse to carry a drunk who has just got off the train. Similarly, as I see it at least, if you were to turn up with your dog (perhaps feigning ignorance of “the rules”), would the driver refuse to take you? I certainly can’t see a situation where he would be forced under the terms of his contract to refuse you, when if he was running a stage carriage service with the self-same bus he would have to take you.
Just some thoughts to throw into the mix