The primary difference would be the lack of spectral bands. That would be pretty easy to pick out, and so I don't think there's an advantage to having a SETI search with this as the primary goal.
For something the size of a plausible ship, the surface area is astronomically smaller, while the temperature of most of its surface would likely be kept at or lower than the range which is compatible with macro-scale organisms and biochemical processes. A ship would also be trying to manage its heat, probably using an insulating exterior.
Meanwhile it would be pointing its thruster - the only part meaningfully above 300K - away from you for at least half the trip if directly oncoming, with angular efficiency coefficients if heading somewhere else. Recall that the ship itself is insulated & temperature-regulated, and therefore shields the emissions of the thruster. The thruster would also likely be focused as much as possible, so we could expect thermal radiation at steeper angles to drop off sharply.
Even something like a nuclear pulse rocket would be hard to spot until it was within a very small neighborhood, and then it would be up to whether you're actually observing that part of the sky during the time when it might be visible.
Ships are very hard to detect until they're almost on top of you.
Thermal insulation isn't necessary or helpful, except as it separates crew spaces from hot spaces and gives the radiators time to reject heat.
Also, the exhaust plume will be very hot and very bright. Even if the ship doesn't radiate, the exhaust will
> That would be pretty easy to pick out, and so I don't think there's an advantage to having a SETI search
So, because you say it's an easy search that's an argument why we shouldn't do it? It's not at all clear that we should have discovered this already by accident, if that's your point. I say it's incredibly hard, and also that it's worthwhile.
Our current telescopes probably aren't good enough to catch a small hot object by now, but there is nothing to gain from explicitly including that as a goal.
Any search which can pick out a small hot interstellar spaceship will be very effective at searching for brown dwarfs and other low temperature objects in space. ("Low" means "not a fusing star".)
I doubt there's anything you can do to optimize a SETI search which makes the identification of brown/red dwarfs more difficult. In fact, since you don't know what the background low-temperature IR sources are, you would need the brown dwarf survey to help characterize what's normal vs. a possible spaceship.
There's already funding for dwarf studies, and it's an area of astronomy which is relatively new and unexplored. (The first brown dwarf was not discovered until 1995, for example.) So we know that's a proposal that many can get behind.
Identification of possible non-stellar objects falls out necessarily and almost trivially (compared to the survey itself) from the data.
Position and velocity data is also important for brown dwarf observations, so you'll get that anyway.
So you can tell me - why should we prioritize a very low probability SETI search when there's a high probability that a brown dwarf survey will be scientifically useful, and end up with useful SETI information?