Yes, I would remove a bit of the soil..., especially as your soil looks HEAVY (i.e., PEATY). Could you lighten the mix by introducing some perlite and COARSE sand? On a positive note, the Dwarf Coconuts are superior for containers as the leaves start dividing up when they are still SMALL.

P.S., What critters are you protecting the palm from??!

I mixed perlite into 3/4 of the soil beneath the surface, adding the final inch or 2 of bagged mix when I thought it was planted too high. Protecting from deer, rabbits, and whatever chews on all my plants! Will remove some soil - thank you!

Better? Or more?

Coconuts fall to the ground and just germinate sitting on top of the soil. There's no need to put any soil around or over them. Their big requirement is heat.

Should be okay at that depth. BTW, I find most any critters really, really are not interested in palms..., you can get rid of the child wire.

LOL, I don't know about rabbits (although I have those too), but I can tell you that palms are not at all desirable to deer. I live in an ACTIVE deer crossing area (I could write the tell all book about deer dietary preferences at this point.) They have never touched any palm or cycad (and actually, a number of other "tropicals"). In the Winter, most of my palms are protected except for the Trachycarpus and a few Sabals and those too are ignored...., even though I have witnessed one munching on a YUCCA in knee-deep snow. Indoors, CATS on the other hand can be fascinated by palms that, I guess, remind them of grass.

I've lost more young palms and other tropicals from either deer, woodchucks, rabbits, chipmunks or whatever. Nothing worse than coming home to see all your palm leaves chewed down and shredded!

Grasshoppers here can play havoc with small coconuts palms but mainly in the dry season when there's not much green pick around. Wet season they seem to have alternatives they prefer and the palms usually put on a lot more than they've lost. But what will totally demolish young coconuts down to the nut are Hermit Crabs. They're only close to the coast so not a worry to most people.