If you go into the LDS site and click on Family History Library catalog, then on place search you can check exactly what films the LDS holds. They have the BTs for Brington, but you seem to be out of luck as they too have a large gap at around the period you are looking for - between 1742 and 1779 only 1755 has survived.
www.genuki.com confirms that the parish register 1754-1812 is missing, and the story for the BTs is as on the LDS site.
You could check to see if there are any Lawrence wills from Brington, but other than that I'm not sure what else you can do. You could also check to see if the overseers accounts and other parish chest entries survived, in the hope that the family might be mentioned.
Also check on
http://www.a2a.org.uk/ to see if there's any mention of them.
Not good news I'm afraid
Regards
David