Given that you are likely booked in P or I class, I would not recommend it.

Unlike Alaska Airlines and United - and Aer Lingus themselves, of course - who give credit for flights in every possible EI business class fare code, BA only gives Avios for the following EI business classes: J, C, D.

I had a quick look on the EI website, and found the following fare on your route for a very similar price to what you've mentioned. It's in P class.

For the specific ticket shown below, these would be the relevant approximate earnings (for the lowest membership status level in each respective programme):

Aer Lingus Aer Club : 4,989 Avios + 250 Tier Credits
Alaska Airlines Mileage Plan: 16,531 Miles
United Airlines Mileage Plus: 7,347 Miles
British Airways Executive Club: 0 Avios + 0 Tier Points