Can you move the shutter in front of the receiver when you know a satellite will transit?

Sure. And for small field of view things that might be the future. But if you are looking at large chunks of the sky at some point with enough satellites the shutter will have to stay shut for the whole duration of the exposure.. so not really a good solution for that.

What if SpaceX funded research into shutters that cover part of the CCD? Is that a possible solution?

Shutters for astronomical instruments are specifically designed to open and close evenly, so that each pixel of the detector receives the same amount of exposure time (as good as is possible). Otherwise it becomes very difficult to calibrate the image, since some parts of the image received more light than others.

Even opening and closing the shutter multiple times during one exposure will cause issues - but probably still possible to get under control. But covering parts of the detector will create significant variants that will be extremely hard to control for. Keep in mind we don't really want pretty pictures, but we use these images to measure the brightness of objects in those images (photometry).

What if spacex just didn't pollute the sky for an entire planet?

We wouldn't get planet-scale internet

(can't reply below) If you had a localised shutter covering pattern X for 6 seconds, could you compensate for that by covering the inverse pattern of X for 6 seconds to get the same exposure for all pixels? So if you covered 3 satellite streaks of 6 seconds each using non-verlapping patterns (A,B,C) you could compensate for that by ending your exposure with covering the inverse of A+B+C for 6 seconds, adding only 6 seconds to your total exposure time and get uniform exposure.

Herein lies the problem. Your proposed solutions are ever more complicated as you begin to grasp the magnitude of the issue. Implementing this idea, and I'm not saying that it's good enough, would require modifications to pretty much every major telescope in the world. And there's no economy of scale because most of them are custom builds. Who's gonna pay for that?

That is only necessary in a small fraction of cases. You can just ignore the part with the streak or lower the shutter when you're using existing equipment.

If the conditions are quite stable that might work (or if one can live with limited accuracy), but conditions tend to change over time (moon position, thin clouds and of course the seeing - how much the atmosphere distorts the image) so nothing is as straightforward as one might hope for.

In space that approach would work quite well indeed, but then you wouldn't need it in the first place..

Edit: Even if one still has to mask the streak in the data in the end, it might still be useful to do, since it will protect the rest of the image from any excess blooming.

How much does the bloom extend in the image fom the streak? Can you measure it in arcminutes or is it optics dependent? Is the bloom present if you cover up the satellite exactly? (Is it caused by the sensor, or would diffraction around a shutter cause it as well?)

I think you could mitigate the problems with conditions changing if (instead of blocking out the streak in the image with some per-area shutter) you tracked the satellite in the image with a horizontal (or vertical) shutter stripe which has the right width to cover the entire satellite. This would make the data lost really minor, and changing conditions wouldn't be a problem, and the exposure would be even. Does that make sense?

If no light falls onto the detector then there is no blooming. And of course there will be diffraction from the shutter edges as well, so engineering it will be quite a challenge (one can take hints from coronographs). If the satellites does end up on the sensor then the dimension of the streak depends on the optics, how bright it is, the used detector, plate scale, etc. (if you are extremely unlucky you might have internal reflections too and they might end up who knows where on the detector - optics is hard).

what if we actually put the astronomical observation satellites above the spacex ones? that should solve the interference problems. Wasn't there a cool new telescope named after James Webb that launched recently?

JWST still hasn't be launched yet.

And space observatories are much much more expensive than ground-based ones. JWST for instance will cost around 10 billion dollars in the end - and they don't even have to pay for the launch cost (one of the contributions of ESA to the project is free launch on Ariane rocket). JWST will have a 8m mirror. Currently three 30m class ground-based observatories are being built (ELT is 39m, TMT 30m and GMT 24.5m) each with an estimated cost between 1 and 1.5 billion dollars. So you get three much much larger ground-based telescopes for the price of one large space observatory (and still have money left over).