Of course.
As an afterthought... even if a route could be copyrighted, we're only talking about the visuals aren't we? After all, his maps tend to be hand drawn illustrations. So if I didn't display the map but instead simply offered a gps file for download - can that be comparible? A gpx file is, after all, simply a list of coordinates. It's something that many people use these days, and something that - so far as I know - he doesn't offer.
If you didn't display his (hand drawn illustration) map then you are not infringing his copyright. You are perfectly entitled to show a drawn trace of the route you took. If you think about it, everyone who publicly shares their tracks and routes on platforms such as ViewRanger or Ordnance Survey would be in bother otherwise.
April makes a good point about the trip reports on here. Are we too all guilty if we happen to write about a walk which just happens to be in one of his books or on his website?
You've done plenty enough research to find out if he has any legitimacy in demanding you sign this agreement and from everything you've posted, no, he doesn't.
Carry on doing what you are already doing and you shouldn't be made to feel like you have to take anything off your site just because the route you took happens to be the same one he wrote about in a book. The route you took is the one you showed on a map that isn't his and you took your own pictures and wrote your own account of it.
Please don't waste any more of your time corresponding with him or his "team".