It shouldn't too hard to get a Bristol/Avon Rider. Last time I had one issued it only took a minute or two searching. You shouldn't have issues with it being accepted on first bus routes, if it is issued on a first bus, nor should you have issue with Abus, as there is many partnerships between the two. I would keep the website open to show proof if the driver is being a bit funny about it!
When I last used the Avon Rider, back in April 2016, The only issue I had was in Bath, trying to board the A4 Air Decker service as far as Keynsham, this is a valid journey on this route, as it states on the website, but the driver was having none of it, saying my ticket price didn't match his machine, even though I told him 5 or more times it was a child ticket. Eventually I jumped off and got the extremely slower First Bus back towards Keynsham and onwards to Bristol... (I think the A4 is only valid on the AvonRider and not Bristol Rider, I had the Avon version.)
I would recommend trying to get your ticket on a bus departing the bus station or where the bus starts/terminates its journey if possible, so the driver does not become late searching for it. If not, it isn't really your problem if they don't know where it is! Just trying to be nice to the staff.
I have not had any other experiences and can't comment on any other operators, I am sure someone else can! Just don't be surprised if some drivers want to take a second look at it.
Have a good time!