I just switched to Nature's Variety a month ago. I too started with the lamb. Murphy had the same reaction. Getting sick a little and burping a lot. I did not finish the bag and switched to Venison. He did fine with venison. I also bought a trial bag of chicken and turkey. He liked it and had no reaction. Today, I tried rabbit. He would not eat it. He kept picking it up and tried hiding the pieces.
I gave him Venison and he was happy.
So, I would switch and try another variety.