Rip the plates off, remove the vin plate, remove the tax disc, and any identifying features at all. Put all the windows through, and tear he seats. Then let the scrappy have it, or a different scrapyard who do not believe in following the law.
When you get your renewal through, SORN it, and keep SORNing it, until your logbook reappears, and then write scrapped on it.
Yet again the Govt (DVLA) are seeking to make money from the motorist, because we are easy targets. �19 for a piece of paper, so you can send it back to them and get rid of the car.
Alternatively sell the car, without the logbook, get written proof of change of ownership, and let the new owner apply for a logbook. You would of course knock the �19 off the sale price.
Incidentally how much are the scrappy offering for your car?