I would take an OTC allergy relief medication since you don't have time to build up your immune system between now and the concert. I understand your issues - I too am a singer in a couple of choirs and have done solo work, so I sympathize with your misery.
I got rid of all my allergy symptoms (seasonal, dust, mold, animal) through a great nutritional supplement that a lot of my friends have had great results with. It's not an overnight fix by any means but it's made a huge difference in my life. I started in November 4 years ago and made it through the winter without bronchitis (which I used to get with every cold, about 3-4 times a winter). By the springtime, I was not affected by the pollen and trees at all and stopped taking allergy meds. It might have been sooner but I just didn't notice it - my symptoms don't hit until April. However, I've also been able to be around cats for the past few years, which I could never do before.
Your animals could be getting warmer winter coats so maybe there is more fur and dander action going on. You could have them groomed and bathed but be sure the groomer doesn't use a really aromatic shampoo that makes you worse.
If your throat is bad, try Chinese hot & sour soup about 2 hours before the concert. Works great. For the nose, get a Neti pot at the drug store or health food store, and use warm water with a pinch of salt and a pinch of baking soda to clean out your sinuses. Feels weird when you do it but it rinses out the junk and is a harmless ancient remedy. The pharmacist can instruct you if the diagrams aren't clear - or email me back for more details.
Sing well!
Edit - just want to add that you can spend the money on an allergy/blood test, but what's that going to do for you? Are you ready to get rid of your animals if you test positive? That's such an emotionally wrenching decision. Limiting contact is difficult when their hair is all over your house anyway. Or would you rather address the problem with a more broad-spectrum approach that will handle any/all allergies?