Hello dumpling, welcome to the forum
I am sorry to hear of your allergies, that must make life very awkward at times, but all is not lost, you dont say, are you vegetarian?
I have successfully lost weight on low carb dieting and I have kept it off too plus it helped me to get my blood sugars down to below 10 (I used to be in the 20s) and eventually to normal non-diabetic levels.
Some people in hear swear that a low GI diet is working for them, so it is up to you to decide which you want to try.
As for your allergies, you can eat other meat, just not poultry? You can also eat soya products, dairy products, cheese, quorn, nuts and if your blood sugars allow you can also eat some pulses and lentils to make sure that you are not lacking in protein, and then of course loads of vegetables, some of the low/medium carb fruit, herbs, spices, seeds, etc and work your diet around that, so just replace chicken or whatever with something that you are not allergic to.
Cut out transfats and only consume healthy fats, from meat, cold pressed oils, eg olive oil, etc butter, get hold of Good Oil (its a hempseed oil and available from bigger supermarket branches), flaxseed oil, so honestly there is no reason to worry about not being able to tolerate fish oils, you still can be really healthy and not have allergic reactions.
All the best
Karen x