I believe that a string bar will probably benefit this build for practical and cosmetic reasons.
That's certainly your choice, but with a strat/tele type of nut, and presuming it's cut properly, the E A D strings shouldn't require a tree or bar even with that distance to the post.

And not to be critical, but your string winding technique (particularly the low E string) could use some improvement.
If you wind the strings so they are a series of single loops going from top to bottom (if that makes sense) it will provide enough break angle right off the nut.

On the low E, I only put 3 wraps on the post. One above the post hole and 2 below. I tend to go 4-5 wraps on my unwound strings, but same over/under method.