I have doubt that adding the Lucas product would help your situation when it appears that you have already diagnosed the problem as "mechanical" in nature. No lubricant or aftermarket additive will fix a mechanical issue. I suppose enough VI improver or higher viscosity oil might reduce some of the problem that you are encountering, but, that might be at the expense of your engine by not using the recommending visosity grade.
I would have second thoughts about using any "aftermarket" oil additive in a vehicle, unless the product is specifically tested for the application and the OEM endorses the product either formally or informally through a forum such as an established racing program. When it comes to additive products, proceed with caution.
My second thought is that you notice black smoke at startup. Please forgive me in advance, but, so what? Does the engine oil require frequent topping off due to the condition? If you are planning on fixing the problem in a few months, then I would suggest that an additional quart or two of oil per month is quite inexpensive when considering the possibility of premature engine damage/failure or other problems encounterd with using the magic pill (oil additives). How would you intend to get the additive out of the engine if you didn't like it? If the "stuff" laid down any deposits, simply draining your oil would not be enough to remove it. Engine replacement would be your only option to get back to square one.
Generally speaking, when it comes to oil additives, don't do it, unless you "actually" know what you are doing and understand the potential consequences. If you know the problem with your car is indeed mechanical, then fix it. Depending on your mileage and the owner's manual recommendations, consider increasing your viscosity one grade (example: 10W30 to 10W40)and see if the smoking or oil consumption decreases until you can fix your ride.
Good luck.