I bathe my cats regularly. It takes awhile for them to get used to it. One tip that helped me is to put a rubber mat in the bottom of the sink so they feel secure and aren't slipping around. The kitchen sink is better because it is smaller and also gives a little more feeling of security. I wrap my left arm around the cat and hold their front paws (for my cats that still struggle a bit). Then I wash with my right hand. I've become very good at grabbing shampoo with one hand while holding the cat firmly with the other.
As far as biting, I've never had one try to bite me while bathing. But they do have little soft muzzles you can buy.
I can empathize with your allergies. I have allergies too and 5 persian cats and bathing them does help a lot. I try not to take so many allergy meds anymore because of other health problems I've had in the last year so. And I figure why should I take uneeded drugs if I can handle it in a more natural way! Wipes and shampoos you don't rinse out just don't do the trick for allergies.
Good luck with your kitties! You can do it if you just be patient and persistant. I have a page on my website on bathing and grooming. It's geared more for thick, long fur but you may find some helpful hints there!