Warning, I am evangelical about Classical Homeopathy. 
There are many alternatives to enhancing a child's health and allergies/asthma status. We use classical homeopathy and this has basically cured my husband's asthma. Dh was on benedryl, proventil
every four hours and claritin and advair daily, due to allergy induced asthma. He is highly allergic to all air borne pollen: all tree, grass, weed, mold, dust, mites, cat dander, pollen, etc. He couldn't be outside for about 6-9 months of the year due to red, runny eyes and sneezing and wheezing and debilitating asthma.
Eliminating these allergens is important, especially in the bedroom. Obviously, some of them can't be eliminated and he would have asthma issues each spring and fall. However, since being on classical homeopathic remedy for four years he hasn't had any asthma in over three years. None. And he no longer takes the benedryl or proventil. Now he never has these issues and only uses Claritin sporadically and the Advair daily. And no more asthma!
We use large doses of vitamin C and Sambucol (for Kids or Adults) which has echinecea (some asthmatics have problems with echinecea). Hydration and humidification (with an aseptically clean humidifier) help to decrease mucus production during colds/flus. A key issue is to identify any dairy or soy allergies and eliminate them. Because these allergens combine to cause severe mucus production when the immune system is under assault of colds/flus.
I have not researched NAET or TAT; but I have heard that they have also been found to help allergies, and subsequent asthma.
You might try the new nasal wash X-clear. It only has water, xylitol, saline and grapefruit seed extract. The xylitol is a natural antibacterial, and the grapefruit seed extract is a natural anti-inflammatory.
I assume that you are keeping the windows and door to the bedroom closed to keep out allergens and consider a HEPA air filter. Eucyliptus helps to open the nasal passages too. Vit. C decreases runny noses, in large doses.
1. Decrease allergens, especially food allergens. (dairy, soy, wheat, eggs are the most common)
2. Keep bedroom allergen free with HEPA filter. (no pets, limited stuffed animals, door closed at all times, plastic cover on mattress, vacuum daily)
3. Vitamin C mega doses (I can't proscribe, of course) I give ds 1000mg, 3x day with first sniffles.
4. Sambucol for Kids (intensive dose, tsp 4x/da) caution with asthmatics some are allergic to echinecea.
5. Consider classical homeopathy, NAET, or TAT as alternative therapies. (Homeopathy cured my husband and friend's allergies/asthma and resolved many of our son's food intolerances.)
6. Avoid steroids like the plague.
7. (We are a totally non-vaccinating family, thumbs down to flu shots)
8. Humidifier, scrupulously clean in his room for sniffles. Clean daily.
9. Avoid antibiotics like the plague.
10. Avoid benedryl-dries mucus membranes
11. Avoid allergy shots
Oh, and dh was severely allergic to cats, we only had one outdoor cat and now we have 5 indoor cats which sleep on our bed!
AND, my friend was on serious drugs for migraines for years, until classical homeopathy. And another friend's daughter was scheduled for tubes to be inserted in her ears until classical homeopathy finally cured what 5 rounds of antibiotics couldn't. She was just 8 months old.
End of PSA for classical homeopathy.
Could you make the cat an "outdoor only" cat? We have one of those due to male territorial issues. He sleeps in the garage, but could have a cat house for cold weather.
HTH,
Pat
