If you only have a D50-relative XYZ/RGB/Lab value for a patch of your ColorChecker, you can approximate the appearance of that patch under another illuminant (e.g., D65) using chromatic adaptation. The linear Bradford adaptation is one way to do this, as you've mentioned. The closer the spectral characteristics of the source and destination illuminations, the better the approximation. Adapting to D65 works reasonably well, but less well when adapting to, say, illuminant A.
Ideally, you would not only have a D50-relative XYZ/RGB/Lab value for your patch, but spectral information: specifically, spectral reflectance samples across the visible range -- roughly 380 nm to 730 nm. From the spectral reflectance data of the patch, the spectrum of an illuminant (e.g., A or D65, or one that you have measured), and the CIE standard observer function data (usually 2-degree), you can very accurately compute the XYZ coordinates, and hence the Lab values relative to that illuminant -- no chromatic adaptation needed.
If you're using a device like an Eye-One Pro to measure your ColorChecker, you can save out the spectral data (not just the derived Lab values).