I don't think it's possible.
RAW to JPEG conversion is very complex process. Like sharpening, saturation you mentioned, any other processing that Lightroom does might be involved as well, such as white balance, ton adjustment, noise reduction, skin smoothing, etc. Also since RAW image has much larger information per pixel (e.g. 14bit) , there's the decision of how to map larger bit depth to JPEG's 8 bit depth.
These processes are very vendor specific. If you want to get the same conversion results, you at least need to know the parameters for each of these vendor specific processes. Unfortunately, raw->jpeg conversion (aka. color rendition process) is the secret sauce of why some cameras seem to render better looking pictures than others (Think about Canon vs Sony, or Apple vs Samsung). So don't expect the parameters of the conversion process will be published by the vendor.
Pleasing color rendition is both important and difficult. First it needs to emulate the response of human eyes and brains to get a "faithful" picture, which is defined by human judgment not sensor's light response. Second, it also try to modify the picture to make it better looking than it really is. Both need to be done under thousands of difference scenarios, which may require different formula for adjustment. I heard Nikon camera stores information of tens of thousands reference scenes (Think about skies, trees, beaches, and all other common or uncommon scenes), and the formula/parameters for tuning those scenes to their best. That's why a same plant picture might look a bit green-ish on Sony/Leica camera, but more natural out of Nikon, even when they use the same sensor. For sure, Lightroom/Adobe has its own secret (very good and possibly similarly complex) formula for automatic processing RAWs, but it cannot possible duplicate the complex RAW process of other vendors.
Also, shooting RAW/JPEG pair isn't your intended solution as well, since JPEG already truncated the pixel information to 8 bit.
Still, you question is very reasonable request and I too have wished there's a such option. I think the best solution is that, in addition to RAW and JPEG, the camera would provide a third format, which let's call Post-Processed RAW (aka, PPRAW). This PPRAW format picture looks exactly like JPEG, because all RAW-JPEG processing has been applied, except it is still the 14 bit format without bit length truncation.
I don't know why Canon, Nikon haven't done this already.