What is the proper salutation for a bishop in an business letter for an invitation?

For a Catholic Bishop or Archbishop you would use:

His Excellency John Smith

Bishop of (diocese)

123 Main Street

Anytown, State zip

or

The Most Reverend John Smith

Bishop of (diocese)

123 Main Street

Anytown, State zip

Both are correct. In the letter you write Your Excellency. Never use Dear before a title.