I think the term "mongrel" could be offensive to the owner of a mixed-breed dog if it were used in a pejorative sense for their pet. Sometimes it is used in a way that implies that mixed-breed dogs aren't as good. It's kind of an old-fashioned word, though. "Mutt" is more common today, at least in the US.
Most dog sports registries have more euphemistic designations for mixed-breed dogs, such as "All American/Canadian" etc., or "crossbred," or "mix." It's all in the context, though.
Depending on your story setting, though, it may be the correct term for a group of dogs of uncertain breeds and mixes, or it could be a term someone would use for whatever reason. Who is applying the term to these dogs? Are they someone who even cares about whether or not the term would offend the owner of a mixed breed dog. I doubt the offense caused by referring to a group of mixed-breed dogs as a pack of mongrels would be a big problem, though.
I think the term "mongrel" is sometimes applied to things like languages, cultures, and to people who are mixed in their ancestry, and that can be very offensive, depending on who us using the term and to whom they are applying it.