, you have proven your GPS updates the TM-D700A and sends the packets with push to talk as a squelch tail, it seems.
Following is from my memory:
Assuming you have two sides displayed:
To enable APRS auto beacon, shift PTT control from the APRS frequency side (I use side A, on the left) to the voice use side (side B, on the right) by pressing the control switch of the right hand volume/squelch control one time. The frequency display goes to a smaller font on the left, and your voice frequency display font enlarges. Your PTT will now work on the right hand side only.
Next, go to the APRS soft key menu by pressing the F soft key (Function Switch) which has 3 separate menu views.
On the APRS view, press the soft key under BCON toggle on and off. Ensure BCON is displayed in the top row of the radio display to ensure it is on. Now If the squelch is set properly on the A side (APRS frequency, your radio should now beacon immediately. If not, toggle BCON on or off a couple of times.
Remember that the APRS auto beacon transmits at the rate you have set up in the main APRS menu. Every two or three minutes is plenty enough, depending on your local traffic.