Keep in mind, past about 10 LS, you're pretty much on passives only anyway, thanks to the inverse square law, and the need to not make the system melt its own controls...
And the absolute energy in solar return between 1000LS and 2000LS isn't really going to be that much in terms of a ship, even at 0.5 C.
If it's approaching dead on, it simply gets brighter. It could be mistaken for a nova or supernova. If it's crossing the sensor, it may be moving to fast for the comparator to make it out as a valid signal, presuming it instead two separate short term contacts.
The angle of approach potentially makes the difference between the detection being referred to defense or astrometrics.
Keep in mind: passives detect "in field objects" by their motion. A scope in space is going to get literally thousands of objects as bright or brighter than a capital ship at 2 AU... let alone a limited signature high-STL "planet killing bullet." They do this by one of several means.
Method 1: Blink Comparator. Take two shots of the same field from the same point. Cancel out anything that didn't move. (This can be done by digital subtraction, or by literally blinking two images on one screen back and forth.) No matter the speed, the object coming straight towards the scope is just a dot for most of the trip... and one that might show up as a faint ring, or might be canceled out.
False positives include close but slow and dim objects,
Method 2: Stereooptic Parallax. Take two images at once from different points of the same deep field. ID all the items by matched signatures (usually spectrographic); anything which is different is closer than the parallax measure's limit; with a 1" of arc accuracy, your "distance" is 1 parsec for a 2 AU baseline separation. Anything closer than your spearation and measurement accuracy's limit distance can be distance determined from the "apparent movement" (parallax). This only finds a location for objects, and is calculation intensive. False positives are non-threat objects which may have had sudden brightness changes resulting in anomalous signatures
Method 3: Parallax on blink. Take two canceled blink images. Then run the parallax calcs on just those objects which moved. Gives you a good 3D location and speed plot. Reduces the parallax calcs.
False positives will be few... but false negatives will be many. Objects moving directly at one scope will appear only in one camera, and may be considered noise.
Can also be done with interferometry.
Method 4: long stable exposure. Set up a stable exposure. Anything with apparent motion appears as a streak. Deep field alignment means this is going to pick up anything with significant apparent motion. But it won't give range.
Method 5: long stable vs reference short exposure. Cancel out the deep field by using a short exposure negative as a mask on the long stable exposure. Makes it easier to see the movement, by reducing the noise. Still doesn't give distance, only the apparent motion of the objects detected. If your reference is old enough, however, it can pick up a new object readily, even at low motion.
Intentional False negative obtainable by picking a bright stable referent, and coming in on a direct line from it to target.
False positives can include close small dim objects.
Method 6: Parallax version of method 5. (Method 4 produces too much to calculate by comparison.) Use your masked views, from the two cameras, and parallax calc on all the signature traces. With sufficient separation, you'll get some apparent motion on anything headed for one or the other. This isn't done yet because of the issue of the field itself not being stable from ground based scopes. I've seen it done with bullets, however, in experimental footage from DARPA released to the media... not good enough to take out the bullets, but the sniper only gets ONE burst.
Keep in mind also: short field scopes only have a few degrees field of view. As in, typically under 2°; deep field scopes have fields of view measured in low double digit arc seconds. So, a spherical search needs over 10000 images. (four square degrees per image presumed for this calc gives 10,314... it will be more, actually...) per sweep, and needs to sweep twice to detect motion on passives, tho' red/blue shifting can imply speed. If each sweep image takes 0.1 sec, and you have 100 scopes assigned the task, you can get one every 10 sec. Which will detect those fast moving things you've been on about... but not the slow moving stuff.
Keep in mind also: signal intensity is a function of duration of exposure, distance, and strength of source (including, for passives, distance from actual source, actual source's strength, and detected objects size and albedo in the relevant wavelengths). Signal sensitivity is a function of the sensor, the size of the aperture and (if any) reflectors.
Almost all detection requires apparent motion to not be questionable/confusable. And if you're going to hit a planet after jumping, you need to be far enough out to correct for the jump's inaccuracy, and fast enough to not be stoppable.
The more apparent motion, the better - but that means sensors distant from the potential target, and that means slower response and ranging times, and fewer surveys per unit time.
Major systems ApCon probably has 3 to 6 arrays of 15° scopes in high orbits, and automated onboard blink comparator processing... but those arrays will still need to have at least 234 units to do continuous sky survey... and probably won't be good for more than a few light minutes. (Wide field also means poor angular resolution, and thus short ranges.)
Active sensors have MUCH better chances of detection, better distance discrimination (due to a controlled and calibrated signal strength) can be used for velocity detection with a single ping (tho it still takes two to get accurate course), and much shorter range. Tho 30° FoV actives are much more common than for passives.