Maybe diclofenac?

Even though it doesn't contain dairy, paracetamol for some reason can be a cause of concern for those with dairy allergies. They are processed differently in the body, so it sort of doesn't make sense, but people be peopling and not fitting in boxes. Be careful of your gut with nurofen and other anti inflammatory meds. And they will affect your blood clotting with long term use.

Nurofen is anti inflammatory and Panadol is only a painkiller, so it might work better anyway.

Is aspirin a possibility? Yes it's a blood thinner, and can trigger gut issues, but it IS an effective painkiller. Not very fashionable atm though.
Paracetamol scares me - the gap between an effective dose and a dose that kills your kidneys is quite small.

check out celebrex. It's like ibuprofen but without the gut skinning