How limited? Will you be able to get on almost daily, once a week, never?
If you can get on about once every 3 days to decent speed internet, here's what I think you should do:
1. Pre-write all your secondaries. Have them done before July 1st so all you have to do is fill out the questionnaire and paste your essay into the textbox.
2. Begin to fill all the secondaries as they come in in early July, hope that you have almost all of them done before going on the trip.
3. Put your essay on 2+ flash drives as well as your computer. SERIOUSLY. Keep one flash drive in your carry on and one in your checked in luggage. Keep your laptop on you. When you get there, keep all 3 copies of your essays in different safe places. Losing them would suck. Alternatively, if your internet access is good enough, you can keep all your essays on Google Drive or Dropbox. I don't like that option though, as it makes you twice as dependent on solid internet access.
4. Get on the internet as frequently as possible to check your email for secondaries. It shouldn't take more than 20 minutes to fill one out since you already have everything ready.
Enjoy the trip and do the secondaries as they come in.