Should I choose a purifier with the British Allergy Seal of Approval?

Not always.
The British Allergy Seal of Approval is actually a paid for endorsement that many high street brands rely upon to add an instant credence to their products.
It can be used a general guide that the product uses a filter that will remove allergens etc and therefore could be beneficial for an allergy sufferer. But that itself is no guarantee that the air purifier itself is particularly effective.

Some of the world's best air purifier manufacturers choose not to subscribe.
Many of the world's more professional air purifier brands do not recognise the Seal of Approval as a true testing authority. Manufacturers like Airfree and Blueair for instance already have a host of more significant independent (not paid for) awards and accolades for their equipment from universities, laboratories and testing houses across the world. This type of certification is a much clearer indication that the product you are considering has been proven to genuinely relieve peoples allergy symptoms in real life situations.

In summary
Just because a product has the British Allergy seal of Approval does not necessarily mean that it is any better than any other. The best indicators are  genuine independent test results. That does not mean that units with the seal of approval are not good machines, some are very good indeed. But it is definitely not the only indicator you should use when making your decision.