A second door is not mentioned in the books

In addition to the maps, The Atlas of Middle-earth also contains a description where Fonstad lays out all the textual sources for each map. If we look at the one for this map we'll see that there's no direct mention of the second door.

The residence of Bag End wound from the great green door west into the side of the hill. The door opened onto a hallway that may have been up to fifteen feet wide, judging from one of Tolkien's drawings. The door faced south, with the opening cut steeply into the bank where the path ran east before turning south to the gate. On the porch Bilbo talked with Gandalf, the Dwarves left their instruments, and Frodo set the hiking packs while preparing to depart.

The hall itself served as an entry closet, with hooks for coats and plenty of room to set out the parting gifts Bilbo had left." Past the entrance doors opened "first on one side and then the other. The best rooms were "on the left, going in," for they had windows cut into the bank, some of which overlooked the kitchen and flower gardens west of the open 'Party' field. Among those rooms were the parlour where the Dwarves met with Gandalf and Bilbo, the dining room, a small sitting room where Bilbo and Gandalf talked before The Party, and the study where Frodo spoke with the Sackville-Bagginses, and later with Gandalf. At least the study and the parlour had hearths. Additionally there was a drawing room where Bilbo was "revived," two or more bedrooms, wardrooms, a kitchen, and "cellars, pantries (lots of those). All in all it was a most comfortable residence.
_{The Atlas of Middle-earth - "Hobbiton and Bag End}

Fonstad does refer though to the kitchen gardens, which a footnote attributes to one of the early drafts presented in The History of Middle-earth.

Finally the field to the south of his front door - it was bordered by his kitchen garden on one side and the Hill road on the other - began to be covered with tents and pavilions.
_{The Return of the Shadow - A Long Expected Party - Second Version}

This draft refers to a kitchen garden bordering the doorway and I guess that Fonstad thought the easiest way to reconcile that information with what we knew of Bag End's door was to assume there was a second door.

A second door is not mentioned in the books

In addition to the maps, The Atlas of Middle-earth also contains a description where Fonstad lays out all the textual sources for each map. If we look at the one for this map we'll see that there's no direct mention of the second door.

The residence of Bag End wound from the great green door west into the side of the hill. The door opened onto a hallway that may have been up to fifteen feet wide, judging from one of Tolkien's drawings. The door faced south, with the opening cut steeply into the bank where the path ran east before turning south to the gate. On the porch Bilbo talked with Gandalf, the Dwarves left their instruments, and Frodo set the hiking packs while preparing to depart.

The hall itself served as an entry closet, with hooks for coats and plenty of room to set out the parting gifts Bilbo had left." Past the entrance doors opened "first on one side and then the other. The best rooms were "on the left, going in," for they had windows cut into the bank, some of which overlooked the kitchen and flower gardens west of the open 'Party' field. Among those rooms were the parlour where the Dwarves met with Gandalf and Bilbo, the dining room, a small sitting room where Bilbo and Gandalf talked before The Party, and the study where Frodo spoke with the Sackville-Bagginses, and later with Gandalf. At least the study and the parlour had hearths. Additionally there was a drawing room where Bilbo was "revived," two or more bedrooms, wardrooms, a kitchen, and "cellars, pantries (lots of those). All in all it was a most comfortable residence.
_{The Atlas of Middle-earth - "Hobbiton and Bag End}

But looking at the sources Fonstad cites in her footnotes, I have two possible ideas for why she may have added this second door.

1. One of the early drafts presented in The History of Middle-earth mentions kitchen gardens on the west side of the field. Perhaps Fonstad thought a door on the west side would make sense for easier access to this garden?

   Finally the field to the south of his front door - it was bordered by his kitchen garden on one side and the Hill road on the other - began to be covered with tents and pavilions.
   _{The Return of the Shadow - A Long Expected Party - Second Version}

2. When Tolkien drew the hill, he made a road in front of Bag-End that continues onwards around the west side and out of sight. Fonstad's second door could be her answer to what this road was for.

   
   _{Close up on The Hill: Hobbiton-across-the Water}