If you book before the end of August you can secure an upgrade to First Class on the Heathrow Express for the price of a the second class fare. In addition a round trip will earn 400 avios (200 one-way).
To book you need to visit this web site.
Once you have chosen your date – and you can book for dates beyond 31st Aug – you see this booking page with the fares:
Note the Business First and Express class are the same fare. Check the Business First fare you want and then pay for your ticket. Tickets are either emailed or sent to your phone. You choose. Present a printout or the phone to the agent on the train.
Still a horridly expensive trip, but if you use it anyway, you may as well do it nicely.
You can read my review of the Heathrow Express here.