It's a beautiful tradition to get animals and other things blessed, and by no means unusual. There are blessings for just about anything, from tractors and seeds to salt and stables.
It is very important to me to get my flock blessed- of course, that involves a farm call. ;)
Many people usually only consider St. Francis' love of animals and nature- usually pictured as him preaching to the birds, or with the wolf of Gubbio, etc. But how many people know that of all animal creatures, his greatest love was for "those innocent animals representing Christ to men"? He had the pleasure of the company of at least several young lambs, and would frequently redeem lambs that were destined for slaughter, in memory of the Lamb Who gave His life that we may live. He would bring these lambs to church with him, and they would give due reverence. One knelt bleating before the altar of Mary and joined in singing. It was a beautiful custom of his to salute flocks as he passed by, and when he did so, often rams and ewes as well as little lambs would gambol around him. Once, he found that a sow slaughtered a newborn lamb that the ewe gave birth to during the night. Much grieved, he said something that I never forgot- and I am paraphrasing St. Bonaventure here-"Woe is me, brother lamb, innocent animal representing Christ unto men! Accursed be that evil beast which slew thee- let neither man nor beast eat of its flesh forever." And some time later, the sow died and shriveled up like a piece of wood.
While many attribute these to be stories and legends, but nothing more, I truly believe that all these occurred. None of these things are that unusual for sheep- only these beautiful natural tendencies were greatly magnified. Of course, sheep, particularly in large flock situations, would never run to greet a stranger, and certainly not play around him, especially when mature. But my lambs will leap with joy when they see me coming, run and gambol and pronk about- and even older sheep will too sometimes, when they feel great joy (however sheep experience that). Sheep are attracted to beautiful music and will sometimes bleat when they hear it. I remember a pet ewe lamb I brought into the living room one winter day and we were singing "Hark the Herald Angels Sing" and she would bleat until the song ended. So I can only imagine what the tremendous holiness of a great saint would compel a sheep to do! :)
I picked St. Francis as my flock's patron saint as well as mine- there seems to be no reason why couldn't you choose a saint for a dog.
I don't know what to tell you about the priest, so I think you got good advice here. But do rest assured that there is nothing wrong with blessing animals! We live in a very traditional parish and our priest has no objection to blessing animals any time of the year.
Have you gotten your house blessed? You could certainly ask to get your dog blessed then.
Pax et bonum!