I believe it doesn't get exported because the text below feature isn't supported.
The GD&T frame gets exported as a GD&T feature as shown in the DraftSight view below, but there's no option shown for text below.
Perhaps this is true for AutoCAD (at least up to R2013, which is the latest version that SW exports to), but I don't have AC so I can't attest to this.
You could create a separate note and group them together.
I see you're on SW2016, so having a separate note and grouping it with the frame may not be a workaround for you (if I'm remembering correctly when grouping was added).
The other option I thought of was making the two into a block, but you can't add the frame to a block, so this path is a dead end.
It looks like if you need to export to DWG (or DXF, same issue), then the below text is a separate note. Joy.
Hopefully, someone else in the forum will post a better reply.
Hello again again,
Since SW is loosing data on the export, I suggest you create an Enhancement Request (ER) or an SPR to get this remedied to at least export the below text as a note entity.
To see how to submit an ER, watch the movie in this post: SolidWorks Enhancement Requests
Don't add anything to this post...that won't accomplish creating your ER nor will it further it along.
(I don't know how to create an SPR, but I trust the how-to is in here somewhere.)
I Have submitted a request.