As Gillumhouse said, most of us call ourselves inns and innkeepers from time to time, so if it's not violating any local rules, I give you permission to change ;-)
I've seen more than one big company change names, the same way Kinkos became FedEx Office over a period of years.
For about a year they were Kinkos FedEx, with the word Kinkos larger and on top.
Then they spent a year as FedEx Kinkos, both words equal size and side by side.
Then they became FedEx Office.
So you might do it in steps, over time. Perhaps start off changing your signs, letterheads, website, etc. to have
Deer Crossing Bed & Breakfast
"The Inn at Deer Crossing"
Like the Inn part is just a slogan or something. Then a year or so later go to
The Inn at Deer Crossing
"Deer Crossing Bed & Breakfast"
Then over time, the new name gets larger and the old name gets smaller until it disappears. Certainly, though, that's some trouble and expense to change signage over time, but that seems to be how big companies change their names without losing the familiarity of the old name: they do it slowly over time..