I've never tried LoDo, and haven't eaten bread or bread substitutes for a very long time now.
In your shoes, were I doing a wrap, I might wrap in a couple of sliceds of ham, then wrap it in foil. In this house, we very, very rarely use clingfilm.
I think your comment, "....But if everyone is having nice food I don’t want to feel different....." is suggesting to me teither hat you don't feel your wrap is nice food, or that your friends will be taking sarnies and you want to "join in".
To be honest, I might be trying to deal with the uderlying question.
In your shoes, I'd be thinking about the nicest low carb picnic options I could get in time, and afford.
At the moment, my local Lidl have in frozen lobster. I'd be inclined to think about lobster, with a salad, and maybe make the salad a bit more special by adding in some salad veg/elements I might not usually deal with. If lobster (say) isn't you thing, then, again, Lidl, have single portion fritatta - in a bout 3 options.
I quite often find if I'm doing the salad when others are doing sarnies, they might comment mine looks so much better than just a sarnie.
I know I'm a few more years down the line that you are, but near the outset of my journey I gave myself a good talking to that "this is the new normal". It's not an instant thing, but if my condition or preferences don't fit in with others, then I plough my own furrow and get on with what I need to do.