I think what you would have to do is use the Import from Catalog function and not just import the images into your master catalog.
Note that you may want to consider only using your master catalog and then using keyword or collections to manage your separate clients. Then you no longer have these extra steps. Not only is it extra work but it is counter to Lightroom's workflow philosophy and so you will be continually wrestling with the program to make it work.
I can answer your question literally, exactly what you have asked.
Or I could suggest something else, more elegant, which may address your real needs better.
Let me do the first:
Use the command "export as catalog" for the new delta of the current job of that repeat client. For the first job you do not need this, as the total catalog is just about the portion you want to have in your master catalog.
Then open your master catalog and use the command "import from another catalog", select this exported portion, and you are done. But caveat: now the same image file is referenced by 2 catalogs, which may lead to,problems, e.g. If you move the image within Lightroom.
Alternative suggestion: just work with 1 master catalog and use a collection set per each of your clients, one collection per job underneath. If appropriate, create more collections in that client's set, e.g. only for images he/she bought actually.
So far I have not understood what an individual catalog per client gives you, what a collection (set) would not?
In addition, the other drawback to catalogs by client and a master catalog is that any changes (edits or metadata) made in one catalog are not recognized by the other catalog. Every time you made a change (edit or metadata) in the client catalog, you'd have to update the master catalog (and vice versa).
So I agree with the others that this is a sub-optimal setup, lots of extra work for no obvious benefit, compared to the one catalog approach.
Thank you so much for this response. I can now understand the drawbacks of having multiple catalogs. However, I find that my larger catalogs are slower to launch and back up, so I wonder how it will be when all of the images are in one catalog. Have you experienced this?
A single catalog should not launch much more slowly (maybe a little more slowly) than smaller catalogs, but that difference is still an advantage to the single catalog approach, because in the multiple catalog approach, each time you launch another catalog, you more than make up for the slightly longer catalog launch time of a single catalog. If it's taking a really long time to launch a larger catalog, then there is something else wrong.
Yes, larger catalogs will take longer to backup. I just walk away from the computer and do something else, or have the backup happen overnight.
Seen on a CreativeLive program-
A professional approach is to do all work in one master catalog. (Use collections for each job as advised above.)
After a time ( eg. 6 months, and a client has finished business) Export the job with image files to another catalog as an archive, and remove from the Master catalog.
Best Images for a portfolio could remain in the Master catalog.
If you are a professional who rarely interacts with older pictures because you have finished the work for that client, I would agree this is a possible approach — although I would still claim that leaving all the photos in a single catalog is more efficient in that it is fewer steps (no time spent creating archives, and no need to find the "best images" and exclude them from the archive, and no need to delete photos from the master catalog that have been archived.
If you are not a professional, then I would not advise breaking up your catalog at all. I want to have the photos from 6 years ago available in my one and only catalog.