the amex plantinum card international travel or something else (forget the name of that stuff) program seems provide slightly cheaper price like 100-300 usd off on top of the market price, you can have a try but need a service fee.
Otherwise, I used to just sign up an alert for the routes i am looking for on skyscanner, I did follow a route for about 6 months and I personally did not really find business class had significant changes of the price over the period of time. But for specific airline if you want to follow, then it might have periodic promotions, otherwise, it did not make a great difference booking 6 months in advance or 2 months, still lowest prices showed up 3-4 months before. But the price got higher at the end.

There are sometimes pretty crazy bug price like 600 usd cathy pacific from vietnam to the US, but it has to be a specific route at certain period of time which was quite hard to meet. There are also time the turkish lira to usd exchange rate significantly dropped and I purchased a ticket in Lira instead of usd on TK business class on top of the promotional fare, leading to additional 500usd cheaper price. But I really did not have a good answer, I would rather just stick to award business ticket which I am currently thinking the best way to book business class as the price was quite stable, flexible and cheap, but you just do not earn elite status and miles.