Have you taught this course before? After teaching it several times from Millman/Parker and other materials using Birkhoff's axioms, I suggest you consider using Euclid himself plus Hartshorne's guide, Geometry: Euclid and beyond, which uses a form of Hilbert's axioms.
The problem for me is that real numbers are much more sophisticated than Euclidean geometry, and the Birkhoff approach is thus a bit backwards except for experts like us who know what real numbers are.
When we covered as much of Millman/Parker as we could manage, the most enjoyable part for the class was the section on neutral geometry, which I learned recently was lifted bodily from Euclid Book I.
If you like assuming that every line in the plane is really the real numbers R, what about going the rest of the way and assuming the plane itself is R^2? Then you can use matrices to define rigid motions and do a lot that connects up to their calculus courses.
Moise is more succinct than the 500 pages suggests as I recall, and is an excellent text from a mathematician's standpoint, but very forbidding probably from a student's. I noticed Moise went from 1.4 to 1.9 pounds from 1st edition to third so maybe the first is also 25% shorter.
The old SMSG books in the 1960's were based on Birkhoff's approach, but are not short. They are also available free on the web.
I just looked at the old SMSG book and found the following circular sort of discussion of real numbers: "if you fill in all those other non rational points on the line, you have the real numbers."
Clint McCrory spent several years developing his own course using Birkhoff's approach at UGA, and made it very successful. Here is a link to his course page. The students loved his class at least in its evolved form after a couple years. they especially appreciated the GSP segment at the start. Apparently many students had little geometric intuition and used that to acquire some. Clint apparently never found an appropriate book to use though.
After teaching this course myself from Greenberg, Millman/Parker, Clemens, supplemented by Moise, and the original works of Saccheri, my own Birkhoff axioms, I finally found Euclid and Hartshorne to be my favorite, by a large margin.
But the beauty of the topic is that there is no perfect choice. You will likely enjoy the search for your favorite too. There is a reason however that Euclid has the longevity it has.
In a nutshell, there are two equivalent concepts, similarity and area, that are treated in opposite order in Euclid and Birkhoff. Euclid's theory of equal content, via equidecomposability, in his Book I, allows him to use area to prove the fundamental principle of similarity in Book VI. Birkhoff assumes similarity as an axiom, and area is relatively easy using similarity, e.g. similarity allows one to show that the formula A = (1/2)BH for area of a triangle is independent of choice of base. Euclid himself uses similarity to deduce a general Pythagorean theorem in Prop. VI.31, whose proof many people prefer as "simpler" than Euclid's own area based proof of Pythagoras in Prop.I.47. The problem is that there has, to my knowledge, never been a civilization in which similarity or proportionality developed before the idea of decomposing and reassembling figures. Briefly, congruence, on which equidecomposability is based, is more fundamental than similarity. Hence, although logically either concept can be used to deduce the other, it seems to me at least that the more primitive concept should be placed first in a course.