What I can predict: Photographing will likely not be accurate enough. As you mention in your step 2, you introduce additional color spaces into the equation, which you'd need to take into account.
Typically, you would print a set of color patches with predefined CMYK values onto the target medium. Then color measurements are taken with a spectrophotometer (and the measuring device needs to be suitable for the medium used - a device that works with paper does not necessarily work for ceramic, too). Since the CMYK values we started with are known, a color profile can then be calculated from these and the values taken from the measurements.
You could check companies like Calibrite, if you are interested in how these solutions look like for home users, or X-Rite for more professional solutions.
However - and this is a big however - this will only help if color management is done right at every step of the production process. Even if you could get color samples printed (which would only be a matter of paying for it), and get a high quality color profile from them: The color profile needs to be applied at the very end of the process chain, while printing. It is in this step, that color values are converted from the working color space used to draw the design to the color space of the printer to get accurate results.
Color management is tedious and can become pretty complicated. In the end, if there is no support from
Tina's side for it, there is not much we can do.