It's possible, but not guaranteed, for your printer to match what your client printed. As you suggested, that the client printed out the RGB file on what is likely a CMYK printer, these colors are likely attainable on the printer's equipment. But the ability to generate these vibrant colors isn't the only thing to consider in the overall problem. Color output can be greatly affected by equipment conditions and paper type, so the color the client produced may only be due to their specific environment and, as a result, your print vendor may not be able to guarantee a perfect match.
Your best option here is to explain your situation to your contact at the print company and have them guide you on how to prepare your file so that they can more reliably match your client's expectations.
I absolutely do not recommend making a habit out of this.
This is the sort of situation where there's a lot of room for variability, which creates a lot of opportunities for a final product your client will be unhappy with. This is not a great position to be in for your client, nor is it a great position for your print vendor. It's in your best interest minimize these chances for something to go wrong.
I understand that file size can be an issue when trying to get files to a client, so the "smallest file size" option can be very attractive for getting client approvals. However, this PDF setting is absolutely not appropriate for color proofing in a print environment. Moving forward, you need to start sending your clients the same file that you would be sending to the printer. If these files are too large, you may want to investigate services like wetransfer or google drive. Because the client would be receiving the same document as the printer, they can be assured that the final print output will match the PDF you provided them.
If you would like to continue using the smallest file size option, you need to explain to your clients that the files you send them are not to be used for color approvals, and that you can't guarantee matches. To circumvent this issue, you could invest in a high quality proof printer and calibration devices and start producing physical proofs with guaranteed color for your clients to approve. These approved copies can then be provided to your print vendor to help them match what the client is expecting.