I'm a finish carpenter and have done a ton of beech. European steamed beech usually has a light "pickled" color to it. You can sand it down, wet it and lightly sand to raise the grain to do some tricks with how you see the extremely limited figuring in beech.
I typically mist beech with a penetrating dye stain highly thinned with acetone, this will raise some grain and darken it up a little. If you want a richer darker tone I then go over that with zar controlled penetration stain because it doesn't have a tendency to get blotchy on you. For your color scheme I might use raw umber to spray and Zara "aged bourbon" color formerly known as medium brown walnut. Let that stuff dry for at least 48 hours because if you don't, any ding or pressure will either delaminate your clear coat or it will fog up and look greyish white between the stain and clear.
The big advantage to this method that I use on all woods and difficult colors is that the dye stain step allows you to cover light spots a little thicker and the spray pattern can be used to mimick or enhance grain. It's not the worst thing to sand it off of you were unhappy either.
I hope this helps you In some way. I might sand yours down and try to richen up the depth of color if it were for me. I'm not harping on your work BTW.
