Yes, you can either call them during the UK afternoon, office hours, which may indeed resolve the issue more quickly. Alternatively you can exchange a Direct Message via Twitter, include your PNR and telephone number (valid for about 3 or 4 hours) and they will call you.

If I can make a general point about this, which may be irrelevant in your case, but having spoke about this at length to someone who works for BA's Social Media team, it's best to focus any complaints very, very tightly. Many complaints go on and on, 3 page letters are not unusual, the record is something like 26 pages for a delayed flight. I'm sure this wasn't the case here, but you are more likely to get a positive resolution if you put one paragraph in only, focusing on what can be fixed (excess fees and Avios) and nothing else (such as rude agents), since the agent then has no other place to go when framing a response.