How to deal with a permanent invade

I generally play fill. I figure as long as my team are comfortable in their roles, I can play a few comfort picks in any role, and focus on team plays, and we can win. But here lately, god forbid jungle pops up on my roster. I lock in something tanky like rammus or sejuini, or i go for my eve pick.

And every. Single. Time. The enemy team locks in a jax, or a kha zix, or a lee sin. They are going to invade me level 2. My team is NOT going to help, theres nothing in my kit that allows me to outdamage or escape them early. They then proceed to steal entire sides of my jungle in stride and repeat invade over and over again. I swear there is nothing in this game more frustrating or helpless in feeling than this. I try to ward, try to simply go for camps on the opposite side, but i'm still losing hundreds and hundreds of gold in farm. My ganks are ineffective because i'm so far behind, and simply pathing to a lane risks getting dove onto again. I can't invade his jungle, it's too far and a certain death sentence. What can I do in this situation? Should I just go to top lane and duo top or something? Its happened to me quite a few times especially when i'm playing a champion that needs level six to fight effectively, and i'm really at a loss here.