I feel like you're overthinking this.
No real issues with stability
Since this object was made by an advanced civilization sufficiently advanced for its technology to be indistinguishable from magic even without the fact of it being magic, fuel to keep its position is not at all a concern. Its bigger concern is dealing with all of the fuel it's getting.
This object is in the path of all of the magical energy coming from the sun. It's in the path of all of the light, solar wind, and solar flares coming from the sun. It therefore has all of the energy it needs to do whatever it does, including keeping itself between the sun and the earth in a fairly stable fashion, and warping cosmic winds and solar flares around it. It needs to know where the earth is to manage this, so it would know what directions would be safe to vent its excess energy - I'm thinking basically four directions, in a square arrangement, maybe around a 45 degree offset from Earth.
Of course, it can't take a noticeable portion of the solar energy heading to Earth or we could recognize it was there based on the differences in the apparent energies received by Earth and by the other planets - unless there's one of these things hiding in their L1 spots as well. But even taking an even 0.00001% of the energy from the sun would be pretty huge. And, of course, the amount of magic energy that it's blocking that we know nothing of is entirely up to you - but realistically speaking, you're probably underestimating what it would be by a large margin.
Think about how much energy you would imagine a human would have access to, unfettered by this device, from standing under the sun at noon, a single shaft of power hitting them and the square meter around them. In two dimensions, Earth has 194 million square meters of surface area pointed at the sun. I chose noon to be able to discount the incident angle. But there's also energy lost due to passing through the atmosphere, if that matters, or passing through space between L1 and Earth, if that matters. So we're talking something in excess of 194 million times what a mage could do with one square meter of power.
Except it's more than that, because it didn't show up during our manned space exploration so far, so it's covering the moon, too. Expanding that out to a disc large enough to cover both is probably not appropriate, because there's really no need to cover out of the plane of the moon's orbit. Using a rectangle area, I think it's something over 4 quadrillion times that reference mage.
Such an object would definitely be discovered if/when earth deployed a space craft that intersected one of the beams of diverted power. If those beams were angled above and below the plane of Earth's orbit, it would seem that would be an unlikely circumstance. This is probably a good thing, because those beams would probably be highly destructive streams of overwhelming power.
It would probably also be discovered if we witnessed anything else encountering those beams. We're talking about at least 48 million times the power of the reference mage mentioned above would have, minus a quarter of whatever the device needed to use itself. This would probably destroy anything that it hit from too short of a range, unless it was specifically made to hit it.
But it wouldn't take that. As Ben mentioned, exploring outside of the shadow produced by the device, any astronauts would be exposed to the magic that it normally shields them from. After they noticed that there was magic in space - that there were long-term effects from what they did out there, so it clearly wasn't them just going insane - it would just be a matter of time until they investigated enough to find the answer.
It wouldn't necessarily be identified just because of an unmanned mission to L1 or beyond. Assuming space wizard technology, it could be slightly out of phase with the rest of matter, such that any such devices could pass harmlessly through it. But any life sophisticated enough to harness the magic leaving the shadow of its effects would probably have a noticeable consequence.