Sankam Residency is a good hotel at Belgaum just off the NH4, but don't know about their dog-friendliness.
.::HOTEL SANKAM RESIDENCY, BELGAUM::.* - sankam, belgaum, hotels in belgaum, north karnataka, hospitality, business hotel, sankamthai
I find the contact numbers are
[FONT=Verdana][SIZE=2][COLOR=#800000]Telephone : +91 831
2455838, 4207888[/COLOR][/SIZE][/FONT]
You could stop over at Hotel Mahendra in Satara on NH4 for lunch, but since the doggie is there betetr to try out any of the open-door dhabas on the Pune-Kolhapur stretch.
You could reach Belgaum in 6-8 hours.