Hi Andrew,
What has almost certainly happened is that you find yourself at a disadvantage with an insurance schemearrangement. Many people refer to M & S cover or Sainsbury's cover when it nothing of the sort. It is cover they have arranged on behalf of their customers with an Insurer with a view to obtaining savings in premium and possible cover enhancements compared to invidual approaches to the company.
For commercial reasons, probably requested by supermarkets, it is badged up as Sainsbury's or M & S, etc cover. They can say that they are looking after their customers by using their influence and purchasing power to arrange a scheme which should benefit everyone. There is nothing wrong with the theory.
The underwriters willl look at the scheme as a whole and it's profitability. You say that, in your case, you have not registered a claim. What has almost certainly happened, judging by the increase, is that the scheme claims paid and outstanding has been reviewed by underwriters exceed the premiums received. The underwriters have, therefore, adjusted their rating when looking at the cost implications at the anniversary date.
It would seem that you are paying for the apparant unprofitability of the scheme. Axa have subsequently quoted you far less as they have judged the risk against their main non scheme portfolio of pet business, which would appear to be more profitable to them. On top of that insurers, with personal lines business, offer some inducement to switch their business to the internet as there are longer terms administrative savings.
Axa are a reputable company...........and, no., I don't work for them. Regardless of whether you renew or go elsewhere I would strongly recommend that you check the respective covers to see if they are the same. If not, you can take a balanced view on both.