Why would a portable barbecue warn against using pots on the grill?
Personal experience: I have used pots and pans on this and similar grills with no problems.  My guess would be, if it says not to use them, it's probably as you said - for the non-stick coating.  There is a specific grill grate you can buy for the larger models, and perhaps you're not going to get the most efficient heat transfer by putting it directly on the attached - but that doesn't mean it won't work at all.  Personally, I'd rather have something slightly less efficient than have to carry two grills.

Not sure which model you have (since you said the portable, I assume this red table-top), but the LX can handle pots and pans according to this site: "Grill lid detaches to make room for large pots or pans." As far as I can tell, they're essentially the same grill as far as how they function, just different sizes and features.

Just to be safe, I would probably fire it up in your yard before the trip and try cooking something that you would cook on the trip, like a pot of coffee or a can of beans.  That way you won't have any surprises when you're deep in the woods and don't have a backup!

If you aren't comfortable using it after your test, or for any reason, there are other solutions available that are pretty easy to carry.  You can get a propane burner, they're very small and light and easy to pack.  There are some that sit directly on top of the propane bottle, but I don't recommend them - they are very easy to tip, especially if you have children at the campsite.

Unrelated to your specific question, but related to camping, I always recommend taking a small cast iron pot or dutch oven of some kind on any camping trip.  In a jam (let's say you run out of propane), you can use that on any fire that you build, even directly on the wood/coals/whatever, and a pot is more versatile than a pan because it can be used for things you said, such as boiling water and making coffee as well as your typical browning, etc.