Surely you are not getting recovery conflicts on the master.
You can avoid recovery conflicts by setting
max_standby_streaming_delay=-1
On the hot standby. This might render the hot standby less useful for other purposes, in which case you might need different hot standbys for different purposes.
> Isn't some of the point of having a hot-standby like this to be able to backup from it?
Not for me. I might use one to dump one or a few tables, but if I wanted a full backup I'd use pg_basebackup, not pg_dump.
Cheers,
Jeff