I do not think it is possible as described.

A lot of the stars near us have been measured and recorded. Some all of them will be routinely measured to estimate their relative motions, and use disturbances in that to detect for missing mass. We can resolve a few stars into discs. We have measured the transmission of the atmospheres of planets and estimated their composition.

We are very far from going in person to the nearest star. If we do ever travel to the stars, it seems reasonable that the region of accurately surveyed space will expand as we travel. Parallax measurements will get the best data from the most distant measurements. There will be many local copies of any stellar mapping, so the outlying districts can process the data and look for any outliers. Surveys will routinely look for anything that has changed in brightness or position. If you pick a 'dull-looking' star and build a Dyson Sphere about it, it will immediately attract attention. SETI and others are already looking for that sort of thing. Faking entries in a database would attract much more interest if it is detected.

How long is this isolation intended to last? Our galaxy occasionally yeets 'rogue stars' into the void. These stars are ordinary stars. Their planetary systems were probably disturbed, so there may be no ancient life. Their relative velocity makes them harder to visit than many other similar stars. This would seem to be the ideal place to lie low: boring, hard to get to, going away, and of little interest once it has gone. Hop there, and you have the isolation you want.

I do not think it is practical to hide stars as described.

A lot of the stars near us have been measured and recorded. Soon all of them will be routinely measured to estimate their relative motions, and disturbances in those will be used to detect for missing mass. We can resolve a few stars into discs. We have measured the transmission of the atmospheres of planets and estimated their composition.

We are very far from going in person to the nearest star. If we do ever travel to the stars, it seems reasonable that the region of accurately surveyed space will expand as we travel. Parallax measurements will get the best data from the most distant measurements. There will be many local copies of any stellar mapping, so the outlying districts can process the data and look for any outliers. Surveys will routinely look for anything that has changed in brightness or position. If you pick a 'dull-looking' star and build a Dyson Sphere about it, it will immediately attract attention. SETI and others are already looking for that sort of thing. Faking entries in a database would attract much more interest if it is detected.

How long is this isolation intended to last? Our galaxy occasionally yeets 'rogue stars' into the void. These stars are ordinary stars. Their planetary systems were probably disturbed, so there may be no ancient life. Their relative velocity makes them harder to visit than many other similar stars. This would seem to be the ideal place to lie low: boring, hard to get to, going away, and of little interest once it has gone. Hop there, and you have the isolation you want.