Funny you should mention it. I am just now changing the tires on my BN2. Here's the story.
About 13 years ago I bought a set of four of the replica
Dunlop wheels from Cape International. At that time all they had was 6-inch wide (I believe they now also have a 5.5-inch version). Yes, they were pricey - and even pricier now - but they look so good, are strong, and they're very, very easy to clean. They are true spline-drive wheels so you put them on the same as wire wheels; no adaptors or anything else needed.
On the recommendation of a well-known Austin-Healey restorer I also bought, at the same time, the very tires you are contemplating. I had them mounted in the UK before the wheels/tires were shipped to me. (I would not do that again, but that's a different story.)
The cost of all this was somewhat staggering (with shipping and customs/duty it came out to about $1,000 ... per wheel), but it was 13 years ago and I've mostly recovered by now. I'll never need another set of wheels for the car, and they really look fabulous.
View attachment 81150
However, I have found the Avon tires to be "more tire" than the car needs. They are virtually street-legal race tires, and I don't drive the car hard. While I don't exactly regret buying the Avons, I have just replaced them with a more conventional tire with a more conventional (original) appearance, namely, Michelin XAS in size 180/15 (80 aspect ratio). I bought these from
Longstone, my go-to tire source for vintage cars.
In fact, I would have bought 5.90/15 cross-ply tires as I like the original look and I like the idea of driving the car the way it looked, felt and handled when new. (I've owned and driven a Healey with 5.90/15 cross-ply tires and an MGA with 6.00/15 cross-ply, and I love it.) However, while my local tire shop said that they could mount 5.90 cross-ply tires on the 6-inch wheels, they also said they'd look funny and I probably would not be happy.
If it were not for the 6-inch wheels, I definitely would have bought a taller, narrower tire. The XAS's are a compromise between the 6-inch wheels and my desire for tires with an original profile (tall and narrow).
All of that said, if you're set on better handling and you plan to drive the car pretty hard, the Avons are likely an excellent choice.