UPDATE
I spoke to BA to clarify 'essential purchases' and where my bag should be delivered to - considering it would not get to Dublin before I left. Spoke to the most helpful agent in the Manchester Exec Club call centre (I am BAEC Gold if it makes a difference). He was able to live chat with the BA Baggage Team at LHR. If he called, it would go to a team in India.
Eventually they confirmed my bag was at LHR (no sh*t), and it would take 3-5 days to get through the backlog. With couriering it to a hotel or address, they like it to go to a place you would be there over a 3 day period to allow for delays. As this was not workable, we went about thinking about what airports would be best. On the 30th, I will be passing Seattle, so that seemed like a good option. He checked with the BA Baggage Team who were not too confident. They didn't want to waste my time and were concerned (maybe overly) if my bag was then misplaced in Seattle.
So it was decided to send my bag to a family member back home in London. I did ask if I could collect the bag from LHR when I land, and ironcially they said 'We are unable to store you bag'. The BAEC agent did laugh at that response. When questioning what I can purchase, he clarified on this occasion, it's like for like and what I need for my holiday. The LHR Baggage Team stated I could spend up to £500, which I did question with the BAEC agent. He did state that it isn't his department, but wouldn't see a problem if I went over it due to my length of trip / as long as it was reasonable. We confirmed I could get an electric razor (mine was only £20, so that's what I'd look to buy). I assume I can get a £20 bottle of aftershave too (basically what I have and what I need).
I understand the limit is 1288SDR (or something like that), so circa £1400 as CWS stated, but I can stick to the £500 limit I think / not to cause a problem.
Thank you everyone for your help/advice! Tomorrow will be an unexpected shopping day - the joys....