Depends if you want a fence, or an heirloom ! Get a couple more quotes from straightforward fencing companies - I'm no expert but beech is a hardwood and expensive, fencing panels are usually softwood and cheaper.
After a run in between my poor cats and the local mange ridden foxes, plus a long running battle with my neighbour's brambles, I've decided a heavy duty six foot fence is no longer a luxury but a necessity!
I got a quote from the company who did my neighbours' fence and it came in at 2.2k for 14 boards plus 25ft of trellis . Does this sound right? I'm in SE London.