The coach is generally as good as a taxi for transfer time, but you have to work around departure times and possible arrival delays. Elizabeth line via Farringdon is fine too, you can tap in and out and I think that gives you a reasonable fare.

You can also prebook a taxi - I pay around £120 for a meet and greet service in a nice car with a firm I've used forever, I expect you can do it for less and if there are two of you you'll likely be paying £50+ for National Express with flexibility so it's not a giant leap from there to a more convenient form of transport. It'll take you directly to the hotel too. I'm a great cheapskate, but I find myself increasingly using the taxi option when I'm travelling with bags and Master BR.