The 4.5 may not offer that much more in terms of objective external or terminal ballistics, but it will offer better objective/subjective controllability.
Rent a 4.5 and a 3.8 Compact (it's unlikely that the range will have a 3.8 to rent, in my experience), and use the full-size magazine with the 3.8 Compact. Shoot a scored, timed metric, holding yourself to either scoring as high as possible in time-allowed or shoot for lowest time possible (if the range allows for rapid-fire). I am willing to bet -and I'm not a betting man- that there's going to be not only a measurable, but also significant, favor to the 4.5. As
@OkiePewPew noted, when paired with its human variable, it's going to be a noticeable difference that's going to magnify at-distance, or, alternatively, at increased cadence.
If you're looking for concealed carry, the 4.5 won't be any harder -or easier- to conceal. The additional muzzle length, if anything, should actually offer for more "keel" stability below the waistline. Since above-the-waist conceivability is going to be a wash between these two guns which are the same in that respect, one might as well take advantage of the keel effect.
If all you're going for is that the 3.8 "looks cleaner," I would weigh this decision with arguments of not only ballistic performance, but also measurable shooting performance - those are truly viable points to debate. If on the other hand the piece will be solely a safe-queen or is destined only for hobbyist use, then by all means, go after it.