Everything I've found while researching indicates that the VX5 and VX6 use exactly the same glass. The significant difference in price is due to the 6x magnification range, the Alumina caps (which are excellent, btw) being included with the VX6, and the fact that 6 has to be better than 5.
My ideal scope for a .375 would be very similar to my ideal for my .308, namely something in the 2-10x range with good light gathering capability, simple illuminated reticle, and preferably without side focus. For me, that means a z6i 1.7-10x42, which is well outside your budget (and mine too, at the moment). Meopta has a similar scope at about half the price that gets excellent reviews, but I don't remember the model name, nor have I ever managed to put my hands on one. It's still out of budget, but not as egregiously as the Swaro.
I REALLY like the Trijicon Accupoints and have them in 3-9x40 and 2.5-12.5x42 flavors. The 2.5-12.5 meets most of my wish list, except that it has side focus. They offer a 1-6 as well. One potential issue with them is bolt clearance. You may have trouble with the bolt clearing the huge occular in low rings. It doesn't clear on either my Defiance or my Christensen, so it ended up on the Mesa Summit instead. Any of the Accupoints could be found under budget. Be aware that there are several iterations of the Accupoint depending on the specific model. The 3-9x40 and 2.5-10x56 are older models and have less eye relief and a different illumination design.
I looked at the VX5 2-10 with illumination and compared it to the z6i and a z8i. The Leupold falls a little short of the Swarovskis in low light, but it's really close considering the substantial price difference. I may yet end up with one for the Defiance (currently wearing the 3-9 Accupoint) unless I can talk myself into spending the money on the z6i.
The Leica Amplus line is another to look at. Again, slightly over budget, but very nice. Some of them have fairly short tube lengths, so you may need offset rings to get it on a .375.
FWIW, my .375 has a discontinued Leupold VXR 2-7 on it.