You'll have no bother with a dog on the Old Man, unless you really go 'off piste'.
The other hills in the group are worthwhile, too, Wetherlam, Grey Friars etc - you could do the whole lot in a day, or break it down into two days. Caw, to the south, is worthwhile too - a grand viewpoint.
If the weather is rough, head into Grisedale Forest, or sample the delights of the Duddon Valley.