Look… if you observed the rabbit to the point you say it didn't seem very well and wasn't moving much, the last thing you want to do is "lob it into the neighbors garden" (not very neighborly, is it?) or even into a close-by field since the reason why it wasn't looking well could be that it was infected with tularemia… which can be spread to other animals as well as humans… see here:
Many routes of human exposure to the tularemia bacteria are known to exist. The common routes include inoculation of the skin or mucous membranes with blood or tissue while handling infected animals, the bite of an infected tick, contact with fluids from infected deer flies or ticks, or handling or eating insufficiently cooked rabbit meat. Less common means of spread are drinking contaminated water, inhaling dust from contaminated soil or handling contaminated pelts or paws of animals…(Source: Illinois Department of Public Health).
It's not to be taken lightly since the infection of humans can cause serious consequences.
The most sensible disposal (recognizing the possible differences between the U.K. and here in the U.S.) is to place it in a double garbage bag using both rubber gloves and a shovel) and take it to the dead animal disposal section of your public refuse service. They have a pit lined with lime or other chemical especially for disposal of dead animals. I realize it's a bit of an effort, but the responsibility is, unfortunately, yours...