I understand that your pup is itchy and your veterinarian wished to try Apoquel, but you are concerned about side effects.
You won't gain anything by using Claritin and Benadryl at the same time. They are both antihistamines with similar actions, so using two won't improve efficacy.
I do want you to use a proper dose of whatever antihistamine you choose however.
And you can use a combination of antihistamines and high doses of omega-3 fatty acids to help with allergy symptoms. In combination fatty acids and antihistamines work synergistically, much better than either one alone.
You can try:
1)Benadryl (diphenhydramine only don't use the combination products with acetaminophen or decongestants as they can be toxic for dogs) at a dose of 1mg to 2mg per pound of body weight or one 25mg capsule per 15 to 25 pounds of body weight orally every 8 hours.
OR 2)Claritin (loratidine) at 5mg per 25 pounds of body weight once or twice daily.
OR 3)Hydroxyzine at 1mg per pound of body weight orally every 8 hours.
OR 4) Chlorpheniramine at 4mg to 8mg per dog once or twice daily.
OR 5) Zyrtec (Cetirizine hydrochloride) at 1/2 mg per pound of body weight orally every 24 hours. That would be one 10mg tablet per 20 pounds of body weight. Make sure it is NOT the formulation with a decongestant (such as Zyrtec-D) because dogs cannot tolerate decongestants.
Some dogs do better on one antihistamine rather than another. Give the one you pick a week trial and if it isn't working try another. Be aware antihistamines can cause sleepiness or hyperactivity in some dogs. These side effects do wear off with repeated use.
Omega-3 fatty acids are fish oil products. 3V by DVM or Derm Caps ES are good brand name products. Use the high end of the dosing schedule for your pup's weight, meaning if she is between doses work up to the higher one. I recommend a dose based upon the EPA portion (eicosapentanoic acid) of the supplement as if we do that the rest of the supplement will be properly balanced. Give her 20mg of EPA per pound of body weight per day. For example an 80 pound dog could take 1600mg of EPA per day.
Cool water baths with an oatmeal shampoo or chlorhexiderm shampoo (which is antibacterial and antifungal) and a conditioner with an antihistamine may help. The water rinses off allergens and the cool temperature soothes itchy skin. Do not bathe her for 2 to 3 days before or after applying topical flea control products or the bath will interfere with the product's efficacy.
Another option if the antihistamines and omega 3's aren't enough is a product called Atopica rather than Apoquel. It suppresses the immune system a bit so it decreases allergic symptoms but it doesn't have as many harmful side effects as systemic steroids.
If you are interested discuss this medications with your veterinarian as Atopica is a prescription product.
Another option if you are interested, is trying immunotherapy. Your girl would need to be tested to determine exactly what she is allergic to, and then she is given small amounts of the allergen to build up her tolerance to it, increasing the amount of allergen in the injection/oral drops incrementally so that her immune system no longer responds to it. This isn't a quick fix, it takes time to slowly build up their tolerance and as she develops new allergens things may need to be added, but it is an option.
If it's been a while since her last exam parasites like cheyletiella, demodex or sarcoptes mites should be looked for by your veterinarian as well if she isn't improving as they can lead to very itchy skin.
Please let me know if you have any further questions.