Your solicitor should be your professional advisor on this one. LA searches check for Planning Permssion, Building regs, and local land charges placed upon the property (amongst other things). They don't confirm the land title - the Land Registry does that. So where you may have seen a drawing showing your house on the whole plot (as you understand it) it is possibly for planning permission request when you have to show the proposed house in relation to the siting on the land. It does not confirm the land ownership.
As you say, a claim for adverse possesion is likely to take several months because LR have to undertake a series of checks and possibly a site inspection. Another choice is to sell him the smaller piece of land, together with a sworn statement that you have been maintaining the fenced additional piece of land for 5.5 years, which would go part way to the minimum of 10 years before one can start the process of claiming adverse possession. He would probably seek a discount from the price because he is not getting all of what he though he was buying.
You may have a claim from the solicitor who helped buy the house in the first place - which is probably of little short term comfort to you just now.