In this day and age a normal letter still bears more weight than it does with just a call.

It depends on how the management of complaints is set up with BA.

I would do a phone call, note that you are going to send a letter if nothing happens and make notes about it,

The agent on the phone will fill it but it will be a different department to work through what they get.

assuming that you donot hear back follow this up with a letter referencing to the phone call.

If high enough, you could address it to a higher instance. Notice3 that of course the first thing they do is pass it down, but you might hit somebody wo is serious about getting complaints worked out.

With higher instance I do mean a person working in that chain, not the highest. If nothing happens to this, then you should later go for the highest instance refering the phone call with no reaction, the letter with no reaction etc.

There are few things more laughable as a person who complaints right away to the highest position. It is a different thing if you prove in your letter that you have worked the hierarchy but not gotten any reactions.