Hi, my hubby and I visit YellowStone all the time....and there are only a couple places that dogs are NOT allowed and that is Norris and Mammoth...because of the hot springs.. Dogs are NOT allowed on the boardwalks...however if you are carrying your baby...or putting him/her in a backpack we highly doubt ANYONE will say anything to you about it. I take my babies EVERYWHERE....and have yet to get in trouble for it.
The main reason dogs aren't allowed on the boardwalks isn't necessarily because they attract wild animals...its because there have been several deaths because dogs have gotten loose....stepped off the boardwalks onto very dangerous ground...fallen thru the thin layer of crust...into boiling hot water and owners have died trying to save them. Those areas are clearly marked...so if it were ME...I would get a breathable backpack..and just put your baby in it. If you plan on putting it in a carrier that is NOT fully enclosed...just make sure your baby cannot jump out. Make SURE he/she is tethered somehow so that you don't have to worry about it jumping out or getting loose. That's where the danger lies.
And don't worry about anything else....there is NO way I'd put my baby in doggie daycare...OR leave it in the car.....UNLESS there are some that don't want to get out...and opt to stay in the car at certain areas....then the air conditioner will hopefully be left on...There are so many other areas you can take your baby...It shouldn't affect your visit one bit!!
Just remember to bring lightweight jackets or hoodies....for the evenings...because even if it's very hot during the day...it always cools off at night and it can get chilly. I'd bring a little jacket for your baby too!
Have a great time...and take LOTS of pictures!!!