Hi,
The more you use a product, the more sensitized to it you can become, and it certainly sounds as if your reactions are becoming more intense. Ideally, you should avoid using colours that contain peroxide, and you're right in that it can be in darker tints too.
If you're insistent on continuing to lighten your hair, then your hairstylist is right that with highlights the bleach would have little contact with the skin so would be less likely to cause a reaction. There are also low-peroxide products that might be more suitable for you, as well as being kinder to the hair, but you need to be very careful and make sure that you get skin tested for any products before they're used on you.
A professional salon will always skin test before using new colours on a client, and if there's any reaction, they should refuse to do the colour service. Sometimes clients can find the skin testing an inconvenience, but it's the most responsible thing to do in terms of client health and safety, as I'm sure you understand.
Good luck!